4)
=('
•4)
5(5) ,
&c.,
where, comparing for example the equations for(Z)(l, 1, 2) and (2Z)(1, 1, 1), it will be
observed that in the first case the contacts 1, 2 of the symbol (1, 1, 2) successively
coalesce with the point 1, giving respectively 2(2, 2) and 3(1, 3), the exterior factor
being in each case the barred number, whereas the second case, where the contacts 1, 1
of the symbol (1, 1, 1) are of the same order, we do not consider each of these symbols
separately ^thus obtaining 2(2, 1)+2(1, 2), =4(2, 1)), but the identical symbol is taken
only once, giving 2(2, 1). Thus we have also
(1, 1,1,1, 1)=(1, 1, 1, l)2(2, 1,1, 1).
67. The value of a symbol involving (2), say the symbol (3Z)(2), is connected with
that of y( 3Z • /) ; but as an instance of the correction which is sometimes required I
notice the equation
(2, 1, 1, 1)=K1, 1. l/)U(m2)(m3)+l(»2)(»3)+3(3, 1, 1) + 2(4,1)},
which I have verified by other considerations.
68. We obtain the series of results :
®( :0=1.
( ../)= 2 ,
( ://)= 4 ,
( 7 //)= 4 ,
( ////)=
WHICH SATISFY GIVEN CONDITIONS.
105
) = n+2m2,
( : / ) =2w+4m— 4,
(•//) = 4w+4m— 4,
(///) =4^+2m2;
(I, 2 )(:)
(•/)
(//)
(M7) ( . }
( 7 )
(//)
= a — 3,
= 2a — 6,
= 2a— 3 ;
=2m 2 42mra+fw 2 — 6m— fw+ 8— fa,,
=2m 2 +4mw + n 2 — 6m— 5% + 12 — 3a,
= m 2 +4mw+2w 2 — 3m— 6w+ 8— 3a;
(M) (
7
(/)
= — 4m— 3%— 4 + 3a,
= — 8m— 8n— 4+6a ;
(1 j 1? 2)( . ) _ g m _j_ 9 % _j_3o_j_ a ^2m+ w— 16),
(/ ) =21m+18^30+a(2mf2w— 26) ;
(1? 7 1> 1)^ . ) ==  m 3 {2m 2 w+ m% 2 +fw 3 — 4m 2 — 7»m— fw 2 +^m— §w— 36+a(— 3m— f^+16),
( /) = fm 3 + 2 m 2 n + 2m» 2 4 ^% 3 — 2m 2 — 8 mn —3 n 2 — m — — 3 6 + a( — 3m — 3% f 2 3 ) ;
( :/ ) =2,
(■■//) = 2 ,
(///) =i;
: ) —2 m+ n— 4,
( • /) =2m+2%— 6,
( //) = m+2w— 4 ;
&*)(.) = « — 6 ,
(/) =« 6 ;
( 2,1,1 \ • ) = m 2 + 2mra+^ 2 7m— fw+18fa,
(/) =m 2 H2mw+ w 2 — fm— 7w+18— fa;
v
106
PROFESSOR CAYLEY ON THE CURVES
®0)
=1,
(•)
=2
(//)
=1;
(B,l) ( . }
= w+2m— 6,
(/)
= 2wf m— 6;
(4) ( . }
=1,
(/)
=1;
which are the several cases for the conics which satisfy not more than four conditions,
and
69. For the conics satisfying 5 conditions, we have
(5) =1,
(4.1) 6,
(3.2) = — 9+a,
(3. 1. 1) =fm 2 +2mn+fw 2 — ^w127— fa,
(2.3) = — 4m — in — 6 + 3a,
(2.2.1) = 6m + 6w+54+a(w+w— 15),
(2. 1. 1. 1) = + rtfn + » 2 + \n* — fm 2 — Smn — §n 2 + — 75
+a(— fm— f»+^),
(I, 4) = — 10m — 8n — 5 + 6a,
(T, 1, 3) = — 8m 2 — 12mn— 3w 2 +60m+57w + 36f a(6mf3w— 45),
(1,2,2) =27m+24rc+2723affa 2 ,
(1, 1, 1, 2) = 2 %2 2 +30mw+ J ^ 2 — 189
+ a(m 2 f 2mw f f w 2 — 2 7m —  2 % + ^f^) — f a 2 ,
(1, 1, 1, 1, l)=^ 2 m 4 +fm 3 w+mV+fmw 3 +^w 4 — fm 3 — 5m 2 n— Amrf— \v?
^m*5mn^^+^n+*iZn+150
+ a( — f m 2 — 3m% — f% 2 +  2 %i + — ^f^) + f a 2 .
70. The given point on the curve to which the symbols 1, 2, &c. refer may be a sin
gular point, and in particular it is proper to consider the case where the point is a cusp.
I use in this case an appropriate notation ; a conic which simply passes through a cusp,
in fact meets the curve at the cusp in two points ; and I denote the condition of passing
through the cusp by 1*1 ; similarly, a conic which touches the curve at the cusp, in fact
there meets it in three points, and I denote the condition by 2*1 ; 1*1, 2*1 are thus special
WHICH SATISFY GIVEN CONDITIONS.
107
forms of 1, 2, and the annexed 1 indicates the additional point of intersection arising
ipso facto from the point 1 or 2 being a cusp. Similarly, we should have the symbols
3*1, 4zl, 5x1 ; but it is to be observed that at a cusp of the curve there is no proper
conic having a higher contact than 2*1 ; thus if the symbol contains 3x1, or a fortiori,
if it contain 4*1 or 5al, the number of the conics is in every case =0 ; it is thus only
the cases 1*1 and 2x1 which need to be considered.
71. The several modes of investigation which apply to the case of contact at a given
ordinary point of the curve are applicable to the case of contact at a cusp : we may if
we please employ the functional method ; we have here a functional equation of the fore
going form, — ( / /) m —ri(4n{4m— 6) + m!(4n+2m— 3) + (\ri 2 —±ri) 4 + (m'ri — a')4+ (
which only differ from the corresponding expressions with 1 in that they contain
n\2m— 3, 2n\4m— 6, 4n\4m— G, 4m\2n— 3
in place of
n\2m— 2, 2n\4m— 4, 4n\4m— 4, 4m\2n— 2
repectively, and they lead to the expressions for (1*1, 1,1:), &c., the arbitrary con
stant being in each case properly determined.
= 1 ,
= 2 ,
=4,
= 4,
= 2 ;
K
72. We have
( ^( :: )
(•••/)
(:// )
( 7 //)
( I III)
MDCCCLXVIII.
\rri 2 —\m!)
pri*%rri)
pri 2 —\iri)
PROEESSOE CAYLEY ON THE CURVES
108
a* 1 . i) ( ... )
(:/)
( 7 /)
(///)
(ra, 2) ( . }
( 7 )
(//)
= w+2m— 3,
=2w+4m— 6,
=4%4m— 6,
=4w+2m— 3;
= «4,
=2a— 8,
=2a — 4 ;
(l/sl, 1, 1)^ . ^ = 2m 2 +2mw+fw 2 — 8m— fw+13— fa,
( • /) =2m 2 +4mw+ ?i 2 — 8m— 7w+18— 3a,
(//) = m 2 + 4mw+2w 2 — 4m— 8w+12— 3a;
(1*1, 3)^ . ^ = — 4m— 3w— 5 + 3a,
(/) = — 8m— 8n— 6 + 6a;
(1*1, 1’ 2)( • ) = 4mf 8w+44+a(2m+ n—17),
(/) =20m+16rc+42+a(2mi2rc27);
(1*1, 1, 1, 1)^ . ) == f m 3 _j_2m 2 w+ mw 2 +fw 3 — 5m 2 — 9mw— 2w 2 + 3 %i{^?i— 57+a(— 3m— fwf/
( / )=fm 3 +2m 2 w+2mw 2 +f^ 3 — fm 2 — lOmw— An 2 — J g L m+ J r^ — 54+a(— 3m— 3n+£
(2*!) ( ... )
=1,
( 7 )
=2
( 7/3
=2,
(///)
=1;
(2^.1) ( . )
=2m+ w— 5,
( 7 )
=2m+2w— 6,
(//)
= m+2w— 4;
(2«1, 2) ( }
=a — 7,
(/)
=a — 6 ;
(2*1’ 1 > !)( • ) = m 2 +2m%+fTi 2 7m^+21fa,
(/) =fm 2 +2wmf n 2 — fm— 7w+lS— fa.
WHICH SATISFY GIVEN CONDITIONS.
109
7 3. The remainder of this table, being the part where the symbols ( • ) and ( / ) do
not occur, I present under a somewhat different form as follows : —
=0,
(4*1, 1)
=0,
(3x1, 2)
= 0,
(3^1, 1, 1)
=0,
(2,3)
—(2x1, 3)
=0,
(2,2,1)
— (2*1, 2, 1)
~n — 3,
(2, 1,1,1)
—(2x1, 1,1, 1)
=i(»3)(»4),
(i, 4)
(1^1,4)
=1,
(1,1,3)
— (1*1,1, 3)
=(2*I, 3) + (»3),
(1,2,2)
(Ixi, 2,2)
=3(w3)+*l,
(1,1, 1,2)
(51,1,1, 2)
=(2*1, 1, 2)+{n3)(7i4)+J+2»3 m
(1,1, 1,1,1)
i(lxl, 1,1, 1,1)
=(2*I, 1, 1, 1).
These results relating to a cusp, are useful for the investigations contained in the
Second Memoir.
It will be noticed that the symbols which contain 2*1 are not, like those which contain
2, symmetrical in regard to (m, n) : the interchange of (m, n) would of course imply the
change of a cusp into an inflexion, and would therefore give rise to a new symbol such
as 2/1 ; but I have not thought it necessary to consider the formulae which contain this
new symbol.
Investigations in extension of those o/De Jonquieres in relation to the contacts of a
Curve of the order r with a given curve. — Nos. 74 to 93.
74. De Jonquieres has given a formula for the number of curves O' of the order r
which have with a given curve U m of the mth order t contacts of the orders a, b, c, &c.
respectively, which besides pass through p points distributed at pleasure on the curve
IJ m (this includes the case of contacts of any orders at given points of the curve U m ), and
which moreover satisfy any other — —(a\b\c\&c.)—g> conditions; viz. the num
ber of the curves C r is =gj(a\l)(b +1)(
1 +
112
PROFESSOR CAYLEY ON THE CURVES
(a, b, c, d, e)—{a\V)(b\\)(c\l)(d\V){e\\)
[m— a ] 5
+ [m— a— l] 4 a [D] 1
+ [m a— 2] 3 /3 [D] 2
+[m a — 3] 2 y [D] 3
+[m— a— 4] ! c$ [D] 4
.+ * PI 1
[2«(a+l)(S+l)(«+l)(^+l)
[m a — l ] 4
+[m— a— 2] 3 a' [D]‘
+[rm— a— 3] 2 j3' [D] 2
+[m— a— 4]V [D] 3
.+ ^ [D]‘
3W
+[S«fe(«+l)(J+l)(e+l) ' \rma 2] s
I +[ma3]V [D]’
 f [m— a — 4]’/3" [D] 2
! + / P>]
W
[S«fe(a+l)(5+l)r [ma3] 1 1][*]*
i +[m— a — 4]y" [D] 1 1
1+ • /3"'[D] 2 J
+[2bcde(a+ l)f [ma4] 1 V][>] 4
1+ ‘TM 1 /
— abode [*] 5 .
78. In all these formulae there is, as before, a numerical divisor in the case of any
equalities among the numbers «, b, c, &c. And D denotes, as before, the deficiency, viz.
its value now is l)(w— 2)— z; or observing that the class n is =
m 2 — m— 2eS— 3*, we have or say D=l— =1 + A if
A= — Tfi   tt k .
79. It is to be observed with reference to the applicability of these formulae within
certain limits only, that the formulae are the only formulae which are generally true ;
thus taking the simplest case, that of a single contact a , the only algebraical expression
for the number of the curves C r which have with a given curve U a contact of the order
«, and besides pass through the requisite number ^r(r+3)— a of arbitrary points, is that
given by the formula, viz.
(a) = {a + 1) (r m — a + AD) — ax.
Considering the curve U m and the order r of the curve C r as given, if a has successively
WHICH SATISFY GIVEN CONDITIONS.
113
the values 1, 2, ... up to a limiting value of a, the formula gives the number of the
proper curves C r which have with the given curve U m a contact of the required order a :
beyond this limiting value the formula no longer gives the number of the proper curves
C r which satisfy the required condition, and it thus ceases to be applicable ; but there
is no algebraic function of a which would give the number of the proper curves C r as
well beyond as up to the foregoing limiting value of a.
80. The formulae are applicable provided only the conditions include the conditions
of passing through a sufficient number of arbitrary points ; viz. when the number of
arbitrary points is sufficiently great, it is not possible to satisfy the conditions specially
by means of improper curves C r , being or comprising a pair of coincident curves. Thus
to take a simple example, suppose it is required to find the number of the conics which
touch a given curve t times and besides pass through 5 — t given points: if the number
of the given points be 4 or 3 there is no coincident linepair through the given points,
and therefore no coincident linepair satisfying the given conditions ; if the number of the
given points is =2, then the line joining these points gives a coincident linepair having
at each of its to intersections with the given curve a special contact therewith, that is,
having in \m(m— 1)(to — 2) ways three special contacts with the given curve ; if the number
of the given points is 1 or 0, then in the first case any line whatever through the given
point, and in the second case any line whatever, regarded as a coincident linepair, has
to special contacts with the given curve ; and so in general there is a certain value for
the number of given points, for which value the conditions of contact may be satisfied by
a determinate number of improper curves C r , and for values inferior to it the conditions
may be satisfied by infinite series of improper curves C r . It is by such considerations
as these that De Jonqujeees has determined the minimum value T of the number of
arbitrary points to which the conditions should relate in order that the formulae may
be applicable : I refer for his investigation and results to paragraphs XVII and XVIII of
his memoir. I remark that in the case where the number of improper solutions is
finite, the formula can be corrected so as to give the number of proper solutions by
simply subtracting the number of the improper solutions : but this is not so when the
improper solutions are infinite in number ; the mode of obtaining the approximate
formula is here to be sought in the considerations contained in the first part of the pre
sent Memoir; see in particular ante. Nos. 8, 9 & 10.
81. The expressions for [a), (a, b), & c. may be considered as functions of rm, 1 + A,
and K, and they vanish upon writing therein rm=0, A=0, x=0 ; they are consequently
of the form (rm, A ,z) l \(rm, A, «) 2 + &c., and I represent by [«], [a, b~]. See. the several
terms (rm, A, *)', which are the portions of (a), (a, b), &c. respectively, linear in rm,
A, and z. The terms in question are obtained with great facility ; thus, to fix the ideas,
considering the expressions for (a, b, c, d ), —
1°. To obtain the term in rm, we may at once write D=l, «=0, the expression is
thus reduced to
(afl)(5l l)(cJl)(^ f 1) { [rm— a] 4 b [rm— a— l] 3 a} ,
114
PROFESSOR CAYLEY ON THE CURVES
=( a +i)(a+iX*+ix«i+i)
and the factor in { } being =rm\rm— a— l] 3 , the coefficient of run is
(a+l){b+l){e+l)(d+l)l*Y\\
which is
= (a+lXHl)(«+l)(^+l)(«+l)(«+2X* + 3)
2°. To obtain the term in A, writing rm — 0, z = 0, and observing that
[D]'=A + 1, [D]»=(A+1)A, [D]*=(A+1)A(A1), [D]*=(A + l)A(Al)(A2),
&c. give the terms A, A, — A, +2A, — 6 A, &c. respectively, the term in A is
(a+l)(b+l)c+l)(d+l) r [a 1> . PA
+ [_«_2] > j3. 1
+[— a — 3] J y . —1
+ i. 2
 «(a+lX« + 2X*+3)
+ ft (“ + 2)(a + 3)
+ 7 ( a +3)
+2S
3°. For the term in z, writing rm— 0, D=l, and observing that [z]\ [z] 2 , [«] 3 , [z] 4
give respectively the terms z, —z, 2 z, —6z, this is
~td («+lX5+l)(c+l){[al] 3 + [«2]V }. 1
+Xcd (a+lXfl—l) {[— a— 2] 2 +[— a— 3]'a" } . 1
tbcd{a— 1) { [_ a _3]>+ a "'}. 2
+ abed . —6
where the terms in { } are
— ( a + 1 — 2)(a 3), («+2 a "X« + 3) and (a + 3a'")
that is,
— (^Hl)(«+2)(a+3), (c+6Z+2)(a(3) and — (6fcFd+3)
respectively : whence the whole expression is
Xd (a + V)(b + 1)(;
and we have moreover
(a+lX^ + l)(c+l)(^+l)=(l+a+/3+7 + ^)i
WHICH SATISFY GIVEN CONDITIONS.
115
the other lines are of course expressible in terras of (a, (3, y, S), but as the law of their
formation would then be hidden, I abstain from completing the reduction.
82. The series of formulae is
[a] = ( a\l)rm
+(a+l)aA
[a, b~\=(a{ l)(6+l)( a +l) . . rm
(a+l)(ft+l)f «(«+ 1)1 A
1 
+ r 2%+i)(j+i)p,
— ab
where a=a\b, (3 —ab; and coeff. of z expressed in terms of a , (3 is=a(l \u\(3)— /3.
[«,M]= (a+l)(&+l)(c+lX*+l)(«+2) .‘.m
+(a+l(«+l)(c+l)[ *(*+l)(«+2)  A
(3(a+2)
+ fX c(a+l)(b+l)(c+l)(a+2)
J+^+c+2)(a+l)
1
where a=a\b{c, (3 =ab{acjbc, y=abc ; and the coefficient of z expressed in term&
of a, (3, y is = — a 3 — a 2 /3 — a 2 y— 3a 2 — a/3 — 2a j 2(3 jy.
0, b, c, d]=(a+l)(b+l)(c+l)(d + l) (a + l)(a+2)(a+3) ..rm
(fl+l)(i+l)(c+l)(^+l) r «(«+l)(«+2)(« + 3)l A
— (3 (a+2)(a+3)
— 7 («+3)
+ f + t d(a+l)(»+l)(c+l)(i+l)(*+2)(«+3)J *,
 S cd(c+d+ 2) (a+l)(b+l) (a+3) I
+22 bcd(b+c+d\3)(a + l)
— 6 abed
where a =a+5+c+d, . . }>=abcd.
MDCCCLXVIIX.
116
PROFESSOR CAYLEY ON THE CURVES
[a, b , c, d, e]= (a+l)(b\l){c\l)(d\l)(e\l)(ct,\l)(cc\2)(ct,\?>)(u\i) rm
+ (a+lX^+lX«+lX^+l)(«+l)f <«+i)(«+2X*+3)(«+4);
I — /3 (a+2)(ct+3X«+4)
— 7 (a+3)(a+4)
 — 2£ (af4)
A
[ —6s
+
%b («+i)(»+i)(c+ix^+iX«+i)
+ ( $]■>
{a, b, c ) = [«][i][c]
+ [a][i, <]+[?][«> c]+ c[a, i]
+ [«, 6, c],
{«, d)= HP]P]M
+t[a, b][c, d]
+%][*, c, d]
+ [«, 6, ^
and so on : this is easily verified for (a, b), and without much difficulty for (a, b , c), but
in the succeeding cases the actual verification would be very laborious.
84. The theoretical foundation is as follows. Writing for greater distinctness ( a) m in
place of (a), we have (a) m to denote the number of the curves C r which have with a given
curve U m a contact of the order a, and which besides pass through ^r(r+3 ) — a points.
Let the curve U m be the aggregate of two curves of the orders m, m' respectively, or say
let the curve XT'" be the two curves m, ml, then we have
a functional equation, the solution of which is
(d)m
WHICH SATISFY  GIVEN CONDITIONS.
117
where [_d\ m is a linear function of n, m, z, or, what is the same thing, of m, A, z. I
assume for the moment that when the coefficients are determined [«] m would be found
to have the value =[«].
Similarly, if ( a , b) m denote the number of the curves C r which have with the given
curve U m contacts of the orders a and b respectively, and which besides pass through
^r(r\3)—a—b points, then if the given curve break up into the curves m, m', then we
have
(a, b) m+ml (a, b) m —(a, b) m/ ={(a) m (b) mf } + {(a) ml (b) m },
where { (a) m (b) m i\ is the number of the curves C r which have with m a contact of the
order a and with m! a contact of the order b, and which pass through the \r[r )3) — a — b
points; and the like for \{a) m \{b) m } . Then, not universally, but for values of a and b
which are not too great, the order of the aggregate condition is equal to the product
of the orders of the component conditions ( ante , No. 12), that is, we have
{(«).(»)}=(«).•(*)= HIAU
{ }=(«}•©»=
and thence the functional equation
{a, b) m+ml —(a, b) m —(a, b) ml =[a] m [b] ml +[a] ml [b] m .
But \a] m &c. being linear functions of m. A, z, we have
M m+ml M.+[«U p] m+mJ P1.+P] ml 5
and thence a particular solution of the equation is at once seen to be [a] m p] OT ; the
general solution is therefore
(a, b) m =[a] m [b] m +[a, b\,
where [a, b~] m is an arbitrary linear function of m, A, z. Hence, assuming for the pre
sent that if determined its value would be found to be =[a, 5], we have the required
formula ( a , 5)=[«][5][a, b~\.
The investigation of the expression for (a, b, c) m depends in like manner on the
assumption that we have
{{a)Jb, c)*}={a) m .{b, P,
and so in the succeeding cases ; and we thus, within the limits in which these assumptions
are correct, obtain the series of formulae for (a, 5), (a, b, c )
85. It is to be observed in the investigation of (a, b) that if o—b , the two terms
[a] m p] w and [_a] m i\J)] m become equal, and the equal value must be taken not twice but
only once, that is, the functional equation is
(a, a) m +w— (a, a) m ,=[a] m [a] m „
and the solution, writing \[_a, d] m for the arbitrary linear function, is
(a, a) m =l[a\[a] m ^\[a, d] m ,
in which solution it would appear, by the determination of the arbitrary function, that
[«, a\ has the value obtained from [a, 5] by writing therein b=ci. Writing the equa
tion in the form
(a, «)=ip][a]+£>, a],
s 2
118
PROFESSOR CAYLEY ON THE CURVES
and comparing with the equation for (a, b), we see that [a, b~\ is not to be considered as
acquiring any divisor when b is put —a, but that the divisor is introduced as a divisor
of the whole righthand side of the equation in virtue of the remark as to the divisor
of the functions (a, b), (a, b, c) ... in the case of any equalities between the numbers
(a, b, c . . .). This is generally the case, and the foregoing expressions for [a, b\ [«, b, c],
&c. are thus to be regarded as true without modification even in the case of any equa
lities among the numbers a, b, c . . . .
86. To complete according to the foregoing method the determination of the expres
sions for (a), (a, b), . . , we have to determine the linear functions [a], \_a, 5], See., which
are each of them of the form fm\gA f hz, where (/' g, h) are functions of r and of
a, b, See . ; and I observe that the determination can be effected if we know the values of
(a), (a, b), Sec. in the cases of a unicursal curve without cusps and with a single cusp
respectively. Thus assume that in these two cases respectively we have
(a)=(a\l)(rm—a),
(a) = (a J \)(rm — a) — a.
Writing first A=— 1, z=0, and secondly A = — l,*=l,we have
(a\l)(rm— a) =fm—g,
(a^l)(rm—a)—a=fm—g\h,
whence
f=(a\V)r, g~[a\l)a, h=—a,
giving the foregoing value
[a] = (af l)r?n + (aj l)aA — uz.
Similarly, for two contacts assume that we have in the two cases respectively
(a, b)=(a\l)(b\Vj\rm— a— $] 2 ,
(a, b)={a\l){b\l)\rm—a—bf—{a(b\l){b{a\l)}\rm—a—b — Y\ l .
Starting here from the formula [a, b~] = (a, b) — [«] \b~\ —fm + gA f hz, and writing suc
cessively A = — 1, z— 0, and A=— 1, z— 1, we have
(«+l)(5l)[m— a— bf— {(a\l)(rm— a)} {(b\V)(rm— b)} —fm—g,
( rd
&P P CO
,s ® >
OS rM 03
§ e= a
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g 8 £
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§ * £
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4 v
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X A
4“ P
*S P
I 3
I + .3
: pt °
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+ V® g
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t 11 §
t! I
F p O
i i r P ’C3
. cu o
I C<3 rtf
CD *i
5 +
£ cl
° fl §
QJ ''— /
P P 04
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« pi §
8 b£> °
^.S 08
d £ .8
.5 o w
3 ^ °
8 «
2 ^ 3
£ T3 °
& p 5
P3 P
TP 04 04
§ 3 S
2 i ■?
rP ci 04
H3 H &
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•p 05 ^
"5 p
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8 53 8 8
OO CO O CO
co co cm to
8 8
CO o
i— I CO
o
+ + + I I I I
4 I I I I
g g g g g
CM ^ o' 00 rjl
I I
S £
CM 05
i— I 'cH
+ +
s s
CM CO
rH 0
78a
— [4m(m — 1 ) (m — 2 )] ;
6 ( 1 , 1 , 1 :)
122
PROFESSOR CAYLEY ON THE CURVES
2 ( 2 , 1 , 1 • )
2(3, 1, 1)
2 ( 2 , 2 , 1 )
24(1,1,1,1 •
6 ( 2 , 1 , 1 , 1 )
= (2 m+w) 2 a
42(2m+w)(12m+12w— 14a)+a( — 4m— n— 3a)
— 336 m — 336%}288 + 135^ 2
but I have not sought to further reduce this expression, not knowing the proper form
in which to present it.
92. The question which ought now to be considered is to determine the corrections or
supplements which should be applied to the foregoing expressions (a), (a, b), &c., or to
their equivalents [«], [a][^]+[«, 5], &c. in order to obtain formulae for the cases beyond
the limits within which the present formulae are applicable ; but this I am not in a
position to enter upon. If the extended formulae were obtained, it would of course be
an interesting verification or application of them to deduce from them the complete
series of expressions (1 : (2 .•.) ... (1, 1, 1, 1, 1) for the number of the conics which
satisfy given conditions of contact with a given curve, and besides pass through the
requisite number of given points. It will be recollected that throughout these last
investigations, I have put De Jonquieres’^ = 0 ; that is, I have not considered the case
of the curves C r which (among the conditions satisfied by them) have with the curve U m
contacts of given orders at given points of the curve ; it is probable that the general
formulae containing the number^? admit of extensions and transformations analogous
to the formulae in which jp is put=0, but this is a question which I have not con
sidered.
93. The set of equations («)=[«], (a, 5)=[«][5]j[«, 5], &c., considered irrespectively
of the meaning of the symbols contained therein, gives rise to an analytical question
which is considered in Annex No. 7.
The question of the conics satisfying given conditions of contact is considered from
a different point of view in my Second Memoir above referred to.
Annex No. 1 (referred to in the notice of De Jonquieres’ memoir of 1861). — On the
form of the equation of the curves of a series of given index.
To obtain the general form of the equation of the curves C” of a series of the index
N, it is to be observed that the equation of any such curve is always included in an
equation of the order n in the coordinates, containing linearly and homogeneously
certain parameters a, b, c . . ; this is universally the case, as we may, if we please, take
the parameters ( a , b, c . .) to be the coefficients of the general equation of the order n \
but it is convenient to make use of any linear relations between these coefficients so
as to reduce as far as possible the number of the parameters. Assume that the
number of the parameters is 1, then in order that the curves should form a
WHICH SATISFY GIVEN CONDITIONS.
125
series (that is, satisfy 3) — 1 conditions), we must have a [a— l)fold relation
between the parameters, or, what is the same thing, taking the parameters to be the
coordinates of a point in ^dimensional space, say the parametric point, the point in
question must be situate on a [a— l)fold locus. Moreover, the condition that the curve
shall pass through a given point establishes between the parameters a linear relation
(viz. that expressed by the original equation of the curve regarding the coordinates
therein as belonging to the given point, and therefore as constants) ; that is, when the
curve passes through a given point, the corresponding positions of the parametric point
are given as the intersections of the (a — l)fold locus by an omal onefold locus; the
number of the curves is therefore equal to the number of these intersections, that is, to
the order of the (a>— l)fold locus; or the index of the series being assumed to be =N,
the order of the (&>— l)fold locus must be also =N. That is, the general form of the
equation of the curves C n which form a series of the index N, is that of an equation of
the order n containing linearly and homogeneously the u \ 1 coordinates of a certain
(oo — l)fold locus of the order N. It is only in a particular case, viz. that in which the
(a— l)fold locus is unicursal, that the coordinates of a point of this locus can be ex
pressed as rational and integral functions of the order N of a variable parameter 0 ; and
consequently only in this same case that the equation of the curves C“ of the series of
the index N can be expressed by an equation (#X^ ? V> z) n =0, or y , l) n =0,
rational and integral of the degree N in regard to a variable parameter 0.
If in the general case we regard the coordinates of the parametric point as irrational
functions of a variable parameter 0, then rationalising in regard to 0, we obtain an equa
tion rational of the order N in 0 , but the order in the coordinates instead of being =n,
is equal to a multiple of n , say qn. Such an equation represents not a single curve but
q distinct curves C”, and it is to be observed that if we determine the parameter by sub
stituting therein for the coordinates their values at a given point, then to each of the N
values of the parameter there corresponds a system of q curves, only one of which
passes through the given point, the other q — 1 curves are curves not passing through
the given point, and having no proper connexion with the curves which satisfy this con
dition.
Returning to the proper representation of the series by means of an equation con
taining the coordinates of the parametric point, say an equation (*Xx, V> 1)“=0, in
volving the two coordinates (x, y ), it is to be noticed that forming the derived equation
and eliminating the coordinates of the parametric point, we obtain an equation rational
in the coordinates (x, y), and also rational of the degree N in the differential coefficient
; in fact since the number of curves through any given point (# 0 , y 0 ) is =N, the
differential equation must give this number of directi6ns of passage from the point
{x 0 , y 0 ) to a consecutive point, that is, it must give this number of values of jr, and must
consequently be of the order N in this quantity.
126
PROFESSOR CAYLEY ON THE CURVES
Conversely, if a given differential equation rational in x, y, ^ , and of the degree N in
the lastmentioned quantity admit of an algebraical general integral, the curves re
presented by this integral equation may be taken to be irreducible curves, and this being
so they will be curves of a certain order n forming a series of the index N ; ' whence the
general integral (assumed to be algebraical) is given by an equation of the abovemen
tioned form, viz. an equation rational of a certain order n in the coordinates, and con
taining linearly and homogeneously the coordinates of a variable parametric point
situate on an (at— l)fold locus. The integral equation expressed in the more usual form
of an equation rational of the order N in regard to the parameter or constant of inte
gration, will be in regard to the coordinates of an order equal to a multiple of n , say
=qn, and for any given value of the parameter will represent not a single curve C", but
a system of q such curves : the firstmentioned form is, it is clear, the one to be pre
ferred.
Annex No. 2 (referred to, No. 17). — On the linepairs which pass through three given
points and touch a given conic.
Taking the given points to be the angles of the triangle formed by the lines (#=0,
y= 0, z— 0), we have to find (f, g , h) such that the conic (0, 0, 0, f, g, KJx, y , z) 2 =0,
or, what is the same thing, fyz+gzx{hxy=0, shall reduce itself to a linepair, and shall
touch a given conic (1, 1, 1, X, vjx, y , zf= 0. The condition for a linepair is that
one of the quantities/' g , h shall vanish, viz. it is/yA— 0 ; the condition for the contact
of the two conics is found in the usual manner by equating to zero the discriminant of
the function ~\—{'K\6f)‘ 1 —(y*\()g)‘ i —(v\6h)' l \2(k\6f)(iJi>\6g)(v\6h)={a, A, c, d\6, l) 3
suppose ; the values of
d— 1 — X 2 — g? — v 2 {2xgjV.
Hence considering (/, g , A) as the coordinates of the parametric point, we have the dis
criminant locus a=0, and the contactlocus
a 2 d 2 + 4 Ac 3 '} 4A 3 d — 3AV— Qabcd= 0,
and at the intersection of the two loci, a— 0, A 2 (4A<2— 3c 2 ) = 0, equations breaking up
into the system {a— 0, A=0) twice, and the system a=0, ibd— 3, 1) 3 =0, considered as a cubic equation in a, has the
twofold root a— —a, that is, we have the above relation between (A, B, C, D). Whence
also writing sind 2 , ae—8bd , ad— 125c) 1 = 0,
(#'X® 2 , ab, b 2 , 2ce—Sd 2 , ae—8bd, ad—\2bc) 1 =0,
which intersect in a threefold locus of the order 24 ; it is to be shown that this contains
as part of itself the quadric threefold locus («=0, b=0, 2ce—3d 2 =0) taken three times,
leaving a residual threefold locus of the order 24 — 6, =18.
We may imagine the coordinates a, b, c, d, e expressed as linear functions of any four
coordinates, and so reduce the problem from a problem in 4dimensional space to one in
ordinary 3dimensional space. We have thus a sextic surface, and two quadric surfaces ;
the sextic is a developable surface or torse, having for one of its generating lines the
line a— 0, b—0 , and for the tangent plane along this line the plane a— 0; the two
quadric surfaces meet in a quartic curve passing through the two points (a= 0, b= 0,
3cc — 2d 2 =Q), which are points on the torse ; it is to be shown that each of these points
WHICH SATISFY GIVEN CONDITIONS.
133
count three times among the intersections of the torse with the quartic curve, the
number of the remaining intersections being therefore 24—6, =18 ; and in order thereto
it is to be shown that each of the points in question (a=0, 5 = 0, 3 ce — 2(2 2 =0)is situate
on the nodal line of the torse, and that the quartic curve touches there the sheet which
is not touched by the tangent plane a— 0 ; for this being so the quartic curve touching
one sheet and simply meeting the other sheet meets the torse in three consecutive points,
or the two points of intersection each count three times.
The torse has the cuspidal line
S=ae— 45<73c 2 =0, T=ace\2bcd— ad 2 — b 2 e— c 3 =0,
and the nodal line
6(ac— 5 2 ), 3 {ad— be), ae\2bd—8c 2 , Z(be—cd ), 6(ce— d 2 ) II
a, b , c , d , e
and the equations of the nodal line are satisfied by the values (a= 0, 5 = 0, 3 ce — 2^ 2 =0)
of the coordinates of the points in question. To find the tangent planes at these points,
starting from the equation S 3 — 27T 2 =0 of the torse, taking (A, B, C, D, E) as current
coordinates, and writing
d=Ad a +Bb 4 +C3 c f Dd d fEd e ,
then the equation of the tangent plane is in the first instance given in the form
S 2 BS— 18TBT=0, which writing therein (a= 0, 5=0, ?)Ce — 2d 2 =9) assumes, as it
should do, the form 0 = 0; the lefthand side is in fact found to be 9c 3 (Sce—2d 2 )A.
Proceeding to the second derived equation, this is S 2 B 2 S + 2S(BS) 2 — 18TB 2 T— 18(dT) 2 =0,
or substituting the values of the several terms, the equation is
9c 4 (AE — 4BD f 3C 2 )
+ 3c 2 (eA4dB+QcC) 2
1 8
in all which equations we have See— 2d 2 =0 ; and if to satisfy this equation we write
c:d: e— 2 : 3)3 : 3/3 2 , then the equations of the tangent planes become
/3 3 (A/3 — 8B)+ 8( 3C/3 2 — 4D/3 + 2E) = 0,
( 3CB 2 — 4D/3 + 2E)  (S 7
20/C Is 4
— 36cB
j
 227>H J
+ 10AGJ
+16/A
+ 40aA 
+ 5AF I
'S 3
lOcG
Js®
— 130AH
+ 20aGj
+ 40^A
+ 10yG
rO
 UC 1
S 2
12aF J
+ 20SA
)
+ 40/F
60/B
Is 8
+ 337iB >
I
1
Ox
o
Q>
— 90/H
J
+ 2SG
— 108«H j
's 5
— a C =<
WHICH SATISFY GIVEN CONDITIONS.
137
where the form of the coefficients may be modified by means of the identical equations
(A,H,GXa,h,g)=K,
(H,B,FX „ )=0,
(G,F,CX „ ) = 0,
(A, H, GJJi, b,f)= 0,
(H,B,FI „ )=K,
(G, F, Cl „ )=0,
(A, H, G Is,/, <0 = 0,
(H, B, FX „ 0=0,
(G,F, CX „ )=K.
There is consequently a conic answering to each value of Q given by this equation, or we
have in all 12 conics.
In the case where the given conic breaks up into a pair of lines, or say,
(a, b, c,f, g , hjx, y, z) 2 =2(Xx\[*y+vz)(}?%+(*'y+iJz),
then, writing for shortness
gjv'—gJv, vX'—v'X, XgJ — x'^=X, Y, Z,
we have
(A, B, C, F, G, H) = (X 2 , Y 2 , Z 2 , YZ, ZX, XY).
Substituting these values, but retaining (a, b, c, f, g, h ) as standing for their values
a= 2XX', &c., the equation in 0 is found to contain the cubic factor 2X0 3 — 3Y0 2 +Z,
where it is to be observed that this factor equated to zero determines the values of 6
which correspond to the points of contact with the cuspidal cubic of the tangents from the
point (X, Y, Z), which is the intersection of the lines ’kx\gjy\vz=Q, and X'x{yj'y\v'z=0;
and omitting the cubic factor, the residual equation is found to be
2eX
— 12cY
— 8/X
20gX
10JX
— 407iX
— 20aX
+ 15aY
+ 5hZ
+ aZ
— 12/Y
+ %Y
 86Y
+ 177tY
+ 4&Z
+ 4c Z
+ 7gZ
= 0 ,
where the form of the coefficients may be modified by means of the identical equations
aX+hY+gZ=0,
7iX+JY+/Z=0,
£X+/Y+cZ=0.
The equation is of the 9th order, and there are consequently 9 conics.
138
PROFESSOR CAYLEY ON THE CURVES
Annex No. 6 (referred to, No. 48). — Containing, with the variation referred to in the
text , Zeuthen’s forms for the characteristics of the conics which satisfy four conditions.
:: )= n f 2m,
( .*./)= 2n \4rn,
( ://)=4w + 4m,
( ■ ///)=4w+2m,
( ////)= 2n +
^’ ^(.\) —2 m( m+ 3) + r,
(: /) = 2m( m+2%— 5) + 2r,
( • //) =2w(2m+ 5)42n— 26)r],
( • /) =^[(m[w)(— (m+w) 2 — 7(mfw) + 48) }4mw(3m3w— 13)2(3 to 13#— 20)(&+r)],
(//) =^[— m 3 + 6mw 2 l2M 3 l13m 2 — \8mn— 30n 2 — 42mj84w+(3ml6w— 26)S] ;
(1? 1? 1> 1)^ . ) = i{ 2(m— 3)(m— 4)(% 2 — m— n)\(n— 3){n— 4)(m 2 — m— n)
+ 4( m 2_ll m+ 28)r +2(^ 2 — 11^H28)S
+ (4(ra  4){m  4)  l)(2i + r) + 2S 2 + r 2 } ,
(/) =i{ (m—3)(m—4)(n 2 —m—n) +2(n—3)(n—4)(m 2 —m—n)
+ 2(m 2 llm+28)r +4(w 2 llw+28)S
+ (4 (n  4)(m  4)  l)(i + 2r) + S 2 + 2r 2 } ;
(2) ('.)= ^ + ;,
(:/)=2(3w+i),
( ' //)— 2(3m+/),
(///)= 3m+ <;
^ (:)=3(2mwJw 2 +4m— 10w){(2m[w— 14)«,
( • /) —2(3m4i)(m\n— 12) +24(mjw),
(//) = 3(m 2 +2mw— 10m\4n)\(mi~2n 14) i
(2, 1, 1)( . ) = (2m+^7)(6r+(w3»
+ {{m — n)[m + n — 5) + r){3m + / — 36)
+ 12(m— n){m\n— 3),
(/) = ( m+ 2w7)(6H(m3»
+ ((n — m)(m f n — 5) + &)(3m f / — 36)
+ 12{n— m)(m+n— 3);
WHICH SATISFY GIVEN CONDITIONS.
139
( 2 ’ 2 \ • )=(3m+/) a 3(3»i+/)9r 8i,
( / )=i(3m+i) 2 3(3m+/)8r9^
: )= 6w— 4m+3;s= 5m— 3w + 3/,
( • /)=10w— 8m+6«=10m— 8^+6/,
(//)= 5w— 3m+3«= 6m— 4w+3/;
( lj 3 \ • )= 2(4m 2 + 3mw+3w 2 +28m32w) + 3(2m+ »— 13)*,
( / )= 2( 3m 2 + 3mw— 4m 2 — 32m + 28w) + 3( m+2w— 13);;
• )=10w— 10m+6»= 8m— 8 w+6/,
(/)= 8w— 8m+6*=10m— 10w+6/.
Annex No. 7 (referred to, No. 93).
In connexion with De Jonquieres’ formula, I have been led to consider the following
question.
Given a set of equations :
a = a (viz.b = b , c —c, See.),
ab = ab /viz. ac= ac Se c., and the like in all the subsequent equationsx
+(ll)a.i\ +(11 )a.c, /
abc = abc
+ ( 12 )(a .bc+b . ac+c . ab)
+(111) a. b, c,
abcd= abed
+ ( 13)(a . bcd\ Sec.)
+( 22)(a5 . cd\ See.)
+ (112)(a ,b.cd\ &c.)
+ 1111 a.b .c .d,
and so on indefinitely (where the ( • ) is used to denote multiplication, and ab, abc Sec.,
and also ab, abc Sec. are so many separate and distinct symbols not expressible in terms
of a, b, c Sec., a, b, c Sec.), then we have conversely a set of equations
a = a (viz. b =b, c=c&c.,
ab = ab /via. ac= ac Sec., and the like in all the subsequent equations^
+ [ll]a . b \ +[ll]a . c, )'
abc = abc
= +[12 ](a.bc+b . ac + c. ab)
+ [lll]a . b . c,
mdccclxviii. x
140
PROFESSOR CAYLEY ON THE CURVES
abcd= abed
+ [13](a . bcd+ &c.)
+ [22](ab.cd+ &c.)
+ [112](a.b.cd+&c.)
+[1111] a.b.c.d,
and so on; and it is required to find the relation between the coefficients ( ) and [ ] ;
we find, for example,
[11]=
(11),
[12]=
(12),
[Hl]= 3
(11)(12)

(HI),
[13]=
(13),
[22]=
(22),
[112]= 2
(13)(12)
+
(22)(11)

(H2),
[1111]= 12
(13)(12)(11)
+ 4
(13)(1H)
 3
(22)(11)(11)
+ 6
(112)(H)

(1111).
And it is to be noticed that, conversely, the coefficients ( ) are given in terms of the
coefficients [ ] by the like equations with the very same numerical coefficients ; in fact
from the last set of equations, this is at once seen to be the case as far as (112); and
for the next term (1111) we have
(1111)= + I2[13][12][11]
= (12) — 12+12 =)— 12
[13][12][11]
 4[1S]{3[12][11][111]}
+ 4
[13][111]
+ 3[22][11][11]
+ (36= ) 3
[22][11][11]
 6[11] 2[1S][12]
+ 6
[11 2] [11]
+[22][11]

[HU]
— [mi]l[112] J

having the same coefficients —12, +4,
— 3, +6, —1 as in the formula for [1111]
terms of the coefficients ( ) ; it is easy to infer that the property hold goods generally.
To explain the law for the expression of the coefficients of either set in terms of the
other set, 1 consider, for example, the case where the sum of the numbers in the ( ), or
WHICH SATISFY GIVEN CONDITIONS.
141
[] is =5 ; and I form a kind of tree as follows:
the formation of which is obvious ; and I derive from it in the manner about to be
explained the expressions for the coefficients [14], [23] See. in terms of the corresponding
coefficients in ( ) ; viz. we have
[14]=
(14),
[23]=
(23),
[H3]=
2
(14)(13)
+
(28)(11)

(113),
[122] —
(14)(22)
+
2
(23)(12)

(H2),
[H12]=
6
(14)(13)(12)

3
(14)(22)(11)
+
3
(14)(112)

6
(23)(12)(11)
+
3
(113X12)
+
1
(23)(111)
+
3
(122)(11)

1
(1112),
[11111]= + 6o
(14)(13)(12)(11)
— 20
(14X13XU1)
+ 15
(14)(22)(11)(11)
3°
(14X112)(11)
+
5
(14X1111)
142
PKOFESSOK CAYLEY ON THE CUEVES
+ 30 (23)(12)(11)(11)
10 (23)(111)(11)
30 (113)(12)(11)
+ 10 (113)(111)
15 (122)(11)(11)
+ 10 ( 1112 )( 11 )
 1 (11111).
To form the symbolic parts, we follow each branch of the tree to each point of its
course : thus from the branch 113 we have
(113) belonging to
[113],
( 11 S)( 111 )
[ 11111 ],
(113X12)
[ 1112 ],
(113)(12)(11) „
[mu];
viz. (113) belongs to [113]; (113)(111), read 11(3 replaced by)lll, belongs to [11111];
(113)(12), read 11(3 replaced by) 12, belongs to 1112 ; (113)(12)(11), read 11(3 replaced
by) 1(2 replaced by) 11 , belongs to [ 11111 ].
And observe that where (as, for example, with the symbol 122) there are branches
derived from two or more figures, we pursue each such branch separately, and also all
or any of them simultaneously to every point in the course of such branch or branches ;
thus for the branch 122 we have
( 122 )
belonging to [ 122 ],
( 122 )( 11 )
( 122 )( 11 )
• (same twice)
»’ [ 1 H 2 ],
( 122 )( 11 )( 11 )
„ [ 11111 ].
Similarly for the branch 23 we have
(23)
belonging to [23],
(23)(111)
„ [ 1 H 2 ],
(23)(12)
„ ’[ 122 ],
(23)(12)(11) (same as infra )
[ 1112 ],
(23)(11)(111)
„ [ 11111 ],
(23)(11)(12) (same as supra)
[ 1 H 2 ],
(23)(11)(12)(11)
[ 11111 ].
We thus obtain the symbolic parts of the several expressions for [14], [23] [11111]
respectively : the sign of each term is + or — according as the number of factors in ( )
WHICH SATISFY GIVEN CONDITIONS.
143
is even or odd; thus in the expression for [11111], the term (14)(13)(12)(11) has foul
factors, and is therefore + , the term (113)(12)(11) has three factors, and is therefore — .
The numerical coefficients are obtained as follows. There is a common factor derived
from the expression in [ ] on the lefthand side of the equation; viz. for [11111], which
contains five equal symbols, this factor is 1.2.3.4.5, =120; for [1112], which con
tains three equal symbols, it is 1.2.3, =6; and so on (for a symbol such as [11222]
containing two equal symbols, and three equal symbols, the factor would be 1.2. 1.2. 3,
=12, and so in other similar cases). In any term on the righthand side of the equa
tion, we must for a factor such as (11), which contains two equal symbols, multiply
by for a factor such as (111), which contains three equal symbols, multiply by
and so on. And in the case where a term (as, for example, the term (122)(1 1) or
(23)(12)(11), vide supra) occurs more than once, the term is to be taken account of each
time that it occurs ; or, what is the same thing, since the coefficient obtained as above is
the same for each occurrence, the coefficient obtained as above is to be multiplied by the
number of the occurrences of the term. For example, taking in order the several terms
of the expression for [1112], the common factor is =6, and the several coefficients are
6, 6.1, 6.1, 6.1x2, 6.1, 6.1, 6. 11x2, 6.1;
and similarly in the expression for [11111] the common factor is 120, and the coeffi
cients taken in order are
120.1, 120.1, 120.1.1.1, &c„
without there being in this case any coefficient with a factor arising from the plural
occurrence of the term.
The foregoing result was established by induction, and I have not attempted a general
proof.
I observe by way of a convenient numerical verification, that in each equation the
sum of the coefficients (taken with their proper signs) is (— ) M_1 1.2 . . (n — 1); if n be
the number of parts in the [ ] (n= 5 for [11111], =4 for [1112] &c.), and moreover,
that the sum of these sums each multiplied by the proper polynomial coefficient and the
whole increased by unity is =0 ; viz. for
[14] [23] [113] [122] [1112] [11111],
the sums of the coefficients are
— 1, —1, +2, +2, —6, +24 respectively,
and we have
l+5(— 1)+10(— 1)+10(2) + 15(2)+10(— 6)+l(24), =7575, =0.
If we have any five distinct things (a, b, c, d, e), then the polynomial coefficients 5, 10,
10, 15, 10, 1 denote respectively the number of ways in which these can be partitioned
in the forms 14, 23, 113, 122, 1112, 11111 respectively, and the lastmentioned theorem
is thus a theorem in the Partition of Numbers.
[ 145 ]
V. Second Memoir on the Curves which satisfy given conditions; the Principle of
Correspondence. By Professor Cayley, F.B.S.
Received April 18, — Read May 2 , 1867.
In the present Memoir I reproduce with additional developments the theory established
in my paper “ On the Correspondence of two points on a Curve ” (London Math. Society,
No. VII., April 1866); and I endeavour to apply it to the determination of the number
of the conics which satisfy given conditions ; viz. these are conditions of contact with a
given curve, or they may include arbitrary conditions Z, 2Z, &c. If, for a moment, we
consider the more general question where the Principle is to be applied to finding the
number of the curves O' of the order r, which satisfy given conditions of contact with a
given curve, there are here two kinds of special solutions ; viz., we may have proper
curves C'  touching (specially) the given curve at a cusp or cusps thereof, and we may
have improper curves, that is, curves which break up into two or more curves of inferior
orders. In the case where the curves C r are lines, there is only the first kind of special
solution, where the sought for lines touch at a cusp or cusps. But in the case to which
the Memoir chiefly relates, where the curves C r are conics, we have the two kinds of
special solutions, viz., proper conics touching at a cusp or cusps, and conics which are
linepairs or pointpairs. In the application of the Principle to determining the numbei
of the conics which satisfy any given conditions, I introduce into the equation a term
called the “Supplement” (denoted by the abbreviation “ Supp.”), to include the special
solutions of both kinds. The expression of the Supplement should in every case be fur
nished by the theory ; and this being known, we should then have an equation leading
to the number of the conics which properly satisfy the prescribed conditions ; but in thus
finding the expression of the Supplements, there are difficulties which I am unable to
overcome; and I have contented myself with the reverse course, viz., knowing in each
case the number of the proper solutions, I use these results to determine a, posteriori in
each case the expression of the Supplement ; the expression so obtained can in some cases
be accounted for readily enough, and the knowledge of the whole series of them will be
a convenient basis for ulterior investigations.
The Principle of Correspondence for points in a line was established by Chasles in
the paper in the Comptes Rendus, JuneJuly 1864, referred to in my First Memoir; it
is extended to unicursal curves in a paper of the same series, March 1866, “ Sur les
courbes planes ou a double courbure dont les points peuvent se determiner individuelle
ment — Application du Principe de Correspondance dans la theorie de ces courbes,” but
not to the case of a curve of given deficiency D considered in my paper of April 1866
146
PROFESSOR CAYLET’S SECOND MEMOIR ON THE
above referred to. The fundamental theorem in regard to unicursal curves, viz. that in
a curve of the order m with \{m— l){m— 2) double points (nodes or cusps) the coordi
nates (#, y, z ) are proportional to rational and integral functions of a variable parameter
Q , — as a case of a much more general theorem of Riemanns — dates from the year 1857,
but was first explicitly stated by Clebsch in the paper “ Ueber diejenigen ebenen Curven
deren Coordinaten rationale Functionen eines Parameters sind,” Crelle, t. 64 (1864), pp.
4363. See also my paper “ On the Transformation of Plane Curves,” London Mathe
matical Society, No. III., Oct. 1865.
The paragraphs of the present Memoir are numbered consecutively with those of the
First Memoir.
On the Correspondence of two points on a Curve . — Article Nos. 94 to 104.
94. In a unicursal curve the coordinates (%, y, z) of any point thereof are proportional
to rational and integral functions of a variable parameter 0. Hence if two points of
the curve correspond in such wise that to a given position of the first point there corre
spond a' positions of the second point, and to a given position of the second point a
positions of the first point, the number of points which correspond each to itself is
=a For let the two points be determined by their parameters Q , S' respectively,
then to a given value of S there correspond a! values of S', and to a given value of S
there correspond a values of S ; hence the relation between (S, S') is of the form
(S, 1)“(S', l) a '=0; and writing therein S'=S, then for the points which correspond each
to itself, we have an equation (S, l) a+a '— .0, of the order a\a! ; that is, the number of
these points is =afa'.
Hence for a unicursal curve we have a theorem similar to that of M. Chasles’ for a
line, viz. the theorem may be thus stated : —
If two points of a unicursal curve have an (a, a') correspondence, the number of united
points is ■=a\a!. But a unicursal curve is nothing else than a curve with a deficiency
D = 0, and we thence infer —
Theorem. If two points of a curve with deficiency D have an (w, a!) correspondence,
the number of united points is =«}a'(2/M) ; in which theorem 2 k is a coefficient to
be determined.
95. Suppose that the corresponding points are P, P', and imagine that when P is given
the corresponding points P' are the intersections of the given curve by a curve 0 (the
equation of the curve 0 will of course contain the coordinates of P as parameters, for
otherwise the position of P' would not depend upon that of P). I find that if the curve
0 has with the given curve k intersections at the point P, then in the system of points
(P, P') the number of united points is
a= a J
whence in particular if the curve 0 does not pass through the point P, then the number
of united points is = af a', as in the case of a unicursal curve. (I have in the paper of
CURVES WHICH SATISFY GIVEN CONDITIONS.
147
April 1866 above referred to, proved this theorem in the particular case where the Jc
intersections at the point P take place in consequence of the curve 0 having a £tuple
point at P, but have not gone into the more difficult investigation for the case where
the Jc intersections arise wholly or in part from a contact of the curve ©, or any branch
or branches thereof, with the given curve at P.)
96. It is to be observed that the general notion of a united point is as follows : taking
the point P at random on the given curve, the curve 0 has at this point Jc intersections
with the given curve ; the remaining intersections are the corresponding points P' ; if
for a given position of P one or more of the points P' come to coincide with P, that is,
if for the given position of P the curve 0 has at this point more than Jc intersections
with the given curve, then the point in question is a united point.
It might at first sight appear that if for a given position of P a number 2, 3, . . ox j of
the points P' should come to coincide with P, then that the point in question should reckon
for 2, 3 . . . or j (as the case may be) united points : but this is not so. This is perhaps
most easily seen in the case of a unicursal curve ; taking the equation of correspondence
to be (0, 1)“(0', l) a '=0, then we have a\a! united points corresponding to the values of
0 which satisfy the equation (0, l) a (0, 1)“'=0; if this equation has a /tuple root 0= A,
the point P which answers to this value X of the parameter is reckoned as / united points.
But starting from the equation (0, l) — 3x, which is right.
101. Investigation of the number of inflexions. Taking the point P' to be a tangen
tial of P (that is, an intersection of the curve by the tangent at P), the united points are
the inflexions ; and the number of the united points is equal to the number of the in
flexions. The curve 0 is the tangent at P having with the given curve two intersections
at this point ; that is, k=2 ; P' is any one of the m — 2 tangentials of P, that is, a! =m— 2 ;
and P is the point of contact of any one of the n — 2 tangents from P' to the curve, that
is, a =n— 2. Each cusp is (specially) a united point, and counts once, whence the Supple
ment is =». Hence, writing 1 for the number of inflexions, we have
/— (to— 2) — (n— 2)+^=4D ;
CURVES WHICH SATISFY GIVEN CONDITIONS.
149
or substituting for 2D its value expressed in the form n — 2m\2\x, we have
t—3n — 3 m+«,
which is right.
102. For the purpose of the next example it is necessary to present the fundamental
equation under a more general form. The curve 0 may intersect the given curve in a
system of points P', each times, a system of points Q', each # times, &c. in such manner
that the points (P, P'), the points (P, Q'), &c. are pairs of points corresponding to each
other according to distinct laws ; and we shall then have the numbers (a, u, a'), (b, (3, (S'),
&c. corresponding to these pairs respectively, viz. (P, P') are points having an (a, a')
correspondence, and the number of united points is =a; (P, Q') are points having a
(f 3 , /3') correspondence, and the number of united points is =b, and so on. The theo
rem then is
j?(a— a— oi')jg'(b— j3— /3')+ &c. + Supp. = 2&D,
being in fact the most general form of the theorem for the correspondence of two points
on a curve, and that which will be used in all the investigations which follow.
103. Investigation of the number of double tangents. Take P' an intersection of
the curve with a tangent from P to the curve (or, what is the same thing, P, P' cotangen
tials of any point of the curve) : the united points are here the points of contact of the
several double tangents of the curve ; or if r be the number of double tangents, then
the number of united points is =2 r. The curve 0 is the system of the n— 2 tangents
from P to the curve ; each tangent has with the curve a single intersection at P, that is,
Jc—n— 2 ; each tangent besides meets the curve in the point of contact Q' twice, and in
(m — 3) points P'; hence if (a, a, a!) refer to the points (P, Q'), and (2 r, (3 , /3') to the
points (P, P'), we have
2{a— a — a'}f{2r— (3— Supp. =2(n— 2)D.
From the foregoing example the value of a — a — a' is =4D — ». In the case where the
point P is at a cusp, then the n— 2 tangents become the n— 3 tangents from the cusp,
and the tangent at the cusp; hence the curve © meets the given curve in 2(n— 3)f3,
=2n— 3 points, that is, (n— 2)\(n— 1) points; this does not prove (ante, No. 96), but
the fact is, that the cusp counts in the Supplement (n—l) times, and the expression of
the Supplement is =(n— l)z. It is clear that we have (3=j3’=(n—2)(m—3), so that
the equation is
8D — 2k + 2r — 2(n  2)(m 3)+(nl)z=(n 2)2D,
that is,
2r=2(n2)(m3)+(n6)2D+(n+3)z;
or substituting for 2D its value =n — 2m\2\x and reducing, this is
2r=w 2 +8m— lOn— 3%,
which is right.
104. As another example, suppose that the point P on a given curve of the order m
and the point Q on a given curve of the order in! have an (a, a!) correspondence, and let
t 2
150
PEOFESSOE CAYLEY’S SECOND MEMOIE ON THE
it be required to find the class of the curve enveloped by the line PQ. Take an arbi
trary point O, join OQ, and let this meet the curve m in P' ; then (P, P') are points on
the curve m having a (m'a, ma') correspondence ; in fact to a given position of P there
correspond a' positions of Q, and to each of these m positions of P' ; that is, to each posi
tion of P there correspond ma! positions of P' ; and similarly to each position of P there
correspond m'a positions of P. The curve 0 is the system of the lines drawn from each
of the a' positions of Q to the point O, hence the curve 0 does not pass through P, and
we have Jc=0. Therefore the number of the united points (P, P), that is, the number
of the lines PQ which pass through the point O, is =ma' +m'a, or this is the class of
the curve enveloped by PQ.
It is to be noticed that if the two curves are curves in space (plane, or of double
curvature), then the like reasoning shows that the number of the lines PQ which meet
a given line O is =ma' {m'a, that is, the order of the scroll generated by the line PQ is
—ma'\m'a.
Application to the Conics which satisfy given conditions , one at least arbitrary . —
Article Nos. 105 to 111.
105. Passing next to the equations which relate to a conic, we seek for (4Z)(1), the
number of the conics which satisfy any four conditions 4Z and besides touch a given
curve, (3Z)(2) and (3Z)(1, 1), the number of the conics which satisfy three conditions,
and besides have with the given curve a contact of the second order, or (as the case may
be) two contacts of the first order; and so on with the conditions 2Z, Z, and then finally
(5), (4, 1), . . . (1, 1, 1, 1, 1), the numbers of the conics which have with the given curve
a contact of the fifth order, or a contact of the fourth and also of the first order . . ., or
five contacts of the first order.
106. As regards the case (4Z)(1), taking P an arbitrary point of the given curve m,
and for the curve © the system of the conics (4Z)(1) which pass through the given
point P and besides satisfy the four conditions, then the curve © has with the given curve
(4Z)(I) intersections at P, and the points P are the remaining (2m— 1)(4Z)(1) intersec
tions ; in the case of a united point (P, P'), some one of the system of conics becomes a
conic (4Z)(1) ; and the number of the united points is consequently equal to that of
the conics (4Z)(1) ; we have thus the equation
{(4Z)(l)2(2ml)(4Z)(l)} + Supp. (4Z)(I)=(4Z)(1). 2D.
107. It is in the present case easy to find a priori the expression for the Supplement.
1. The system of conics (4Z) contains 2(4Z • ) — (4Z/) pointpairs*; each of these, re
garded as a line, meets the given curve in m points, and each of these points is (specially)
a united point (P, P') ; this gives in the Supplement the term m{2(4Z • ) — (4Z/)}. 2. The
number of the conics (4Z) which can be drawn through a cusp of the given curve is
* The expression a pointpair is regarded as equivalent to and standing for that of a coincident linepair :
see First Memoir, No. 30.
CUEVES WHICH SATISFY GIVEN CONDITIONS.
151
= (4Z • ); and the cusp is in respect of each of these conics a united point; we have
thus the term «(4Z • ), and the Supplement is thus =m{2(4Z • )— (4Z/)} f «(4Z • ).
We have moreover (4Z)(1)=(4Z • ), 2D=w— 2m+2+« ; and substituting these values,
we find
(4Z)(1)= (4m2)(4Z . )
 m{2(4Z . ) — (4Z/)} — *(4Z . )
+(w— 2m+2+;s)(4Z • )
= n{ 4Z • )+m(4Z/),
which is right.
108. It is clear that if, instead of finding as above the expression of the Supplement,
the value of (4Z)(l),=w(4Z ■ )+m(4Z/), had been taken as known, then the equation
would have led to
Supp.(4Z)(l)=m{2(4Z . )(4Z/)}+*(4Z • );
and this, as in fact already remarked, is the course of treatment employed in the re
maining cases. It is to be observed also that the equation may for shortness be written
in the form
(4Z) (1) — 2(2m— l)(l)j
+ Supp. (1)=(1)2D ;
viz. the (4Z) is to be understood as accompanying and forming part of each symbol ;
and the like in other cases.
109. We have the series of equations
(4Z) { (1) — (I)(2m — 1) — (f)(2m— 1)}
+ Supp. (1) =(1)2D;
(3Z) { (2) — (2) (2m — 2) — ( 1 , 1)}
+ Supp. (2) =2(2)2D ;
(3Z) 2{(2)(T,l)(2)(2m2)}_
+ {2(1, 1)(1, l)(2m— 3)— (T, l)(2m— 3)}
+ Supp. (T, 1) =(I, 1)2D;
(2Z) { (3) — (3)(2m— 3) — (1, 2)}
+ Supp. (3) =3(3)2D;
(2Z) 2{(3)(2, 1) — (2, 1)}
+ {(2, 1) — (2, l)(2m4)(T, 1, 1)2}
+ Supp. (2,1) =2(2, 1)2D;
(2Z) 3 {(3) — (I, 2) — (3)(2m — 3) }
+ {(1, 2)— (1, 2)(2m4)(l, 2)(2m— 4)}
+ Supp. (1, 2) =(I, 2)2D ;
152
PROFESSOR CAYLEY’S SECOND MEMOIR ON THE
(2Z) 2{(2, 1)— (T, 1,_1)2 — (2, l)(2m4)}
+ {3(1, 1, 1)(1, 1, l)(2m— 5) — (I, 1, l)(2m— 5)}
+ Supp. (1,1,1)
(Z) { (4) — (4)(2m— 4) — (I, 3)}
+ Supp. (4)
(Z) 2{(4)(3 L l)(22)}
+ {(3, l)(3, l)(2m— 5) — (1, 1, 2)}
+ Supp. (3, 1)
(Z) 3{(4)(2,2)(3,I)}
+ {2(2, 2)— (2, 2) (2m— 5)— (1, 1, 2)}
+ Supp. (2, 2)
(Z) 2 { (3, 1)— (2, 1, 1)2— (2, 1, 1)2}
+ {(2, 1, 1)— (2, 1, l)(2m6)(*I, 1, 1, 1)3}
+ Supp. (2,1, 1)
(Z) 4 { (4) — (1^3) — (4)(2m — 4) f
+ {(1, 3)(T, 3)(2m— 5)— (T, 3)(2m — 5)}
+ Supp. (r, 3)
(Z) 3{(3, 1) — (T, 1, 2) — (3, l)(2m— 5)}
+2{2(2, 2)(I, 1, 2)— (2, 2)(2m— 5)}
+ {2(1, 1, 2)(I, 1, 2)(2m— 6)— (I, 1, 2)(2m6)}
+ Supp. (T, 1, 2)
=(1,1, 1)2D;
=4(4)2D ;
= 3(3, 1)2D ;
= 2(2, 2)2D;
=2(2, 1, 1)2D ;
=(1, 3)2D;
= (I, 1, 2)2D;
(Z) 2{(2, 1, 1)(I, 1, 1, 1)3— (2, 1, l)(2m— 6)}
+ {4(1, 1, 1, 1)— (T, 1, 1, l)(2m 7)(I, 1, 1, l)(2m— 7) }
+ Supp. (T, 1, 1, 1) =(1, 1, 1, 1)2D.
110. I content myself with giving the expressions of only the following supplements.
Supp. (4Z)(1) =m[2( .)"(/)]+*(■ )•
Supp. (3Z)(2) =K 2( : )( • /)]+M • /)•
Supp. (3Z)(1, 1)= ( %nn — %n 2 — n\ncc )( :)
+(2m 2 — imn — 2m+2w+(m— ^)a)(  /)
+ (m 2 +m )(//).
CURVES WHICH SATISFY GIVEN CONDITIONS.
153
Supp.(2Z)(3) =M 2(.*.)— (*/)]
+^[ 2 (..)(:/) + 2 ( 2 (:/)(.//))]
+H/y
Supp. (Z)(4) =a*+6(2*+2/),
where a, 5 are the representatives of the condition Z.
It may be added that we have in general
Supp. (Z)(4X) = »— 1)(2( • 4)(/4j)J
+(»+!)( • 4 )
+ (— m+§)(/4)
+ 5%(5)
Verification is 1 — 4 + 3. 1 = 0.
— 75m— 75%+a( 45)
8m 2 — 20m%— 8m 2 + 104m + 104%+a(6m+6%— 66)
+ 1?%— 4% + a(
1 )
10%m + 8% 2 — m—  3  % + a( — 6% + 10)
8m 2 + 10m%
m—  3 %+a( — 6 m +10)
+ 5%.
122. Ninth equation :
4(4, 1)
+ 2(3, 2)
+ 2(3, 1, 1)
+ JSupp. (1, 1, 3)— *(1, 1, 3)
{_ ( m — 2)(2( . 1, 3) — (/ 1, 3))/
+ (»+2)(l,8)
+ (_ m + 2)(/l,3)
+»(4(4, l)+2(2, 3))
— 32m 2 — 80%m— 32% 2
j — 3m 3 — 20m 2 %~20m% 2 — 3% 3 + 109m 2 +232m%+109% 2
1 ' ' — 2m 2 + 12m%+ 2% 2
+ 8m 2 %+12m% 2 +3% 3 — 16m 2 — 80m%— 59% 2
+ 3m 3 + 12m 2 %+ 8m% 2 — 59m 2 — 80 m%— 16% 2
— 4m%— 4% 2
CURVES WHICH SATISFY GIVEN CONDITIONS.
159
(1)
( 2 )
(3)
f 416m4416M+a(
4240m + 240M+«(
— 868m— 868m+c«(
+24m+24M264)
— 8 m.— 8m— 156) + 6a 2
3 mi 2 + 1 2mM + 3 m 2 — 6 9m — 6 9m + 5 82 ) — 9a 2
(4) — 6m + 30 M+a( — 4 m— 10 m— 6)+3a 2
(5) + 112m+106M+a( — 6mM— 3M 2 +12m+45M— 78)
(6) + 106m + 112M+a(— 3m 2 — Qmn + 45?m+12m— 78)
(7) — 36m + a( + 6 m ).
Verification is ( — 2 + 12f 2)2 — 6 + 30 + 3(( — 410)2 — 6)43.9.2=0.
123. Tenth equation:
= 0 :
3(3, 2)
+( 2 , 2 , 1 )
4jSupp. (I, 2, 2)— js(T, 2, 2) 
{— (m— 2)(2( • 2, 2)— ( / 2, 2))} j
+(»+ 2 )( • 2 , 2 )
+(m + 2)(/2, 2)
+ 3m(3, 2)
4 360m 4 360 m
24m 2 4 54 ?mm 4 24m 2 — 468m — 468 m
+ 6 m 4 33 m
— 27mM— 24m 2 4 54m 4 48m
24m 2 — 27 mn 4 48m 4 54m
 27m
(1)
(2)
(3)
(4)
(5)
(6)
o) + «( — 12 ?m— 12m— 234) + a 2 ( + 9)
(2 ) +a( — 8m— 8M+327)+a 2 (m+M— 12)
(3) +«( — 3m— 13) + a 2 ( 4  1)
( 4 ) 4 a( +20 m— 40)4a 2 ( — \n\ 1)
(s) +a( 20m — 40) + a 2 (^m 4 1)
(6) + «( 4~ 3m ).
Verification is + 33 + 3(— 3.2 — 13)49.2=0.
124. Eleventh equation:
3(3, 1, 1)
+ 4(2, 2, 1)
+ 3(2, 1, 1, 1)
+ Supp.(T,l, l,2)*(r, 1, 1, 2 ) ]
l(mi)(2(.l, 1, 2) — ( / 1, 1, 2))J
+ (»+JX ■ 1. 1. 2)
— fm 3 — 30m 2 M— 30 ?mm 2 — m 3
I + 1 8m 3 + 9 Om 2 M + 9 0 ?mm 2 + 1 8 m 3
— §mi 3 — fm 2 M— 6mim 2 — 3m 3
— 24m 2 M— 36mM 2 — 12m 3
+ (™+f)(/l, 1, 2)
+ m( 3(3, 1 l) + 2(2, 2, 1))
— 1 2m 3 — 3 6?m 2 m — 2 4mM 2
4 m a M+ 6mM 2 + m :
(U
( 2 )
(3)
(4;
US)
,6)
■ 7 )
160
PKOEESSOR CAYLEY’S SECOND MEMOIE ON THE
(1) +^§ m 2 + 348mw+ a  z w 2 — 1302m — 1302w+a( (1
( 2 ) + 96m 2 + 216mw+ 96w 2 — 1872m— 1872w+a( ( 2 )
( 3 ) — 522m 2 — 1044mw— 522w 2 +3960m+3960w+a( ^m 3 +3m 2 w+3wm 2 +^w 3 re?
a) + ^m 2 —  2 %iw+ 18w 2 — 2m— 191w+a( «>
<5) + 56m 2 + 252mw+196w 2 — 392m— 392w+a( — m 2 n—2mn 2 —^ri > (I
(6) + 196m 2 + 252mw+ 56 n 2 — 392m— 392w+a( — ^m 3 — 2m 2 w— mw 2 (6)
(7) — xmw — ^^w 2 + 189w+a( (»
o) fm 2 +18mw+ fw 2 — 873) + a 2 ( —  2 )
(2 )  32m 32w+1308)+a 2 ( 2m + 2w48)
o) — ^m 2 — 78mw— ^n 2 \ 358m+ 358w— 2880)+a 2 (— m— fw+84)
(4) — fmw— 2w 2 + 3%^+ ^f%+ 55) + a 2 ( m+ w—
(5> + m 2 4 ^mwH^^ 2 — i  1 m i T i ^+ 322) + a 2 ( fm — f)
(6) + 3 1 m 2 + 3 %m+ fw 2 — i x%— i %+ 322) + a 2 ( +f%— f)
(7> 2mw+ 2w 2 & 2 % ).
Verification is 4(—  163) + 2(^^+18)2191
+ 3((f2)4 + (¥+ i F)2 + 55) + 9((l + l)4^.2)=0.
125. Twelfth equation :
2 ( 2 , 1 , 1 , 1 )
+5(1, 1, 1, 1, 1)
+ ( Supp. (I, 1, 1, 1, 1)— *(1, 1, 1, 1, 1) ]
] — (m— §)(2( • 1, 1, 1, 1)— (/ 1, 1,1, 1))}
+ (»+t)( • !. 1 . 1 , 1 )
+ (w+t)(/l s 1, 1, 1)
+ 2 n( 2 , 1 , 1 , 1 )
^jm 5 +^m 4 w+^V+m 2 w 3 + 1 ^mw 4 w
(31
mV m 2 n 3 — \mn 4 —^n 3 w
— Yjm 5 — \m 4 n— mV — f m 2 w 3 — ^pun 4 re
(6
o) + 12m 3 + 60m 2 w+ 60mw 2 + 12w 3 o
(2) — j^m 4 — ^m 3 n— lOmW— ^mn 3 — t^ti 4 — ££m 3 — :L ^^m 2 n— 1 ^hnn 2 — ^^n 3 re
(3) _ im 4  m 3 w — £m 2 n 2 — 3 mn 3 — \n 4J r 4m 3 + ^m 2 w+ ^mw 2 f 9w* $:
(4) + fm 4 +fm 3 w+ i 3 1 mV+^mw 3 +fw 4 — fm 3 + ffm 2 w+ ^mw 2 + 1 2  L 4 % 3 w
(5> +^m 4 +^m 3 w+ i 3 1 mV+x^mw 3 + fw 4 + 2 J ^m 3 + ^m 2 n\ *^mn 2 — m 3 re
(6) ^m 3 w+ 2mW + 2wm 3 + ^w 4 — 5 m 2 n— 16m% 2 — 5w 3 re
CURVES WHICH SATISFY GIVEN CONDITIONS.
161
— 348m 2 — 696m%— 348% 2 f 2640m f2640%a( fm 3 f 2 m 2 % + 2m% 2 +f% 3
_ 3^ m 2_ 3% m  iL tt% 2 + ^f%+«( fm 2 %+ 3m% 2 +f% 3
+ ^m%+ ^% 2 — 150%+a( (6)
— 15m 2 — 52 m%— 15% 2 } 1920) +a 2 (— 3m— 3%f56)
+ L  L m 2 +115m%+ L f% 2 — ia /%+2430) + a 2 ( ^m+^%— 75)
— f m 2 — m%+ f% 2 — fm— ~ 3 %— 34)ja 2 (— fm— f%+13)
— 4m 2 — hnn—  4 % 2 + • L pm+ 1 ~ i 2 ^ — 238)fa 2 ( — f%+ 3)
— ^m 2 —  2 m%— 4% 2 + i ff 1 m+ f%— 238)+a 2 ( — fm + 3)
— 3m%— 3% 2 + 29% ).
Verification is
(it¥3i)8+(4+¥+¥+9)4+G¥i4H*)2+(¥+*F)
+ 3((+l + l+i)8+(fl+f)4 + (¥)234) + 9(()4+13.2) = 0.
126. It will be observed that in the eighth and following equations, viz. those
wherein the expression of the Supplement contains the symbol (1), I have included along
with the Supplement within the { }, the terms — (m — f {2( • 4) — (/4)} &c., viz. these
are — (m— f) into number of pointpairs (4), &c. : this is for convenience only; it sim
plifies the calculation, both from the symmetrical form under which the remaining terms
present themselves in the several equations, and because the expressions of the terms
in question, (these terms being mere multiples of a number of pointpairs) are by
Zeuthen’s theory known in terms of the Capitals. It is to be noticed that for any
equation, to find the system to which the Capitals belong, we diminish by unity the
barred number and then remove the bar ; thus for the seventh equation, where we have
Supp. (2, 1, 1, 1), the Capitals belong to the system (1, 1, 1, 1).
162
PROFESSOR CAYLEY’S SECOND MEMOIR ON THE
127. Referring to Nos. 41 to 47 of the First Memoir, for convenience I collect the
capitals which belong to a single curve, giving the values in terms of m, n, a as follows.
( 1 )
( 2 )
( 4 )
( 3 )
( 1 , 1, 1, 1 )
A =£$(&_ 1) —lm 3 + 2 m 2 n  im 2 — 2 mn+ 8rc 2 + \m 2 n
+ <*(§ m 2 +  m 6n+ f)+f,;
B =h(n— 4)(m— 4) = \m 3 n — 2m 3 m 2 w+4mw 2 + 10m 2 — 14mw— 16w 2 — 8m +64
+ «( — f mn+ 6 m+ 6w— 24);
C =r.i(m4)(m— 5)= YmV + 2m 3 im 2 wmw 2 18m 2 + £ww + 5rc 2 + 40m 5
+ a(— fm 2 —15);
D =/.(m— 3)(m— 4) = — fm 3 +^m 2 —18m
+ «( m 2  \m + 6).
(2, 1, 1)
( 3 )
E =&(w4)
=
\rtfn

2m 2 
mw+ 4ft 2 + 2m— 16:
+«(
— fw+ 6);
( 3 )
F =2&(m— 3)
=
m 3

4m 2 +
8m% + 3m— 24
+*(
— 3m +9);
(6)
G = 2r(m— 4)
=
mft 2 +
8m 2 —
m» 4ft 2 — 32m+ 4
+ a(
 3m +12);
(2)
D[=i.£(m3)(m4)
swprai] ;
(1)
H =hz
=
— fm 2 ft
+
fmft— 12ft 2
+ «(
— ^m+^ft )fc;
(2)
I =x(n— 3)(m— 4)
=
— 3mft 2
+
9mft+12ft 2 36i
+*(
mw 3m— 4ft + 12);
( 5 )
J =i(m— 3)
=

3m 2
+ 9m+a(m'
( 2 , 2 )
(9) K = r
(3) L =z(n — 3)
(1) M 1)
^w 2 + 4m— ^+a( — f);
— 3w 2 +9w+a( n—S);
fw 2 +ftt+a( — 3ft— ^) + ^a 2 ;
CURVES WHICH SATISFY GIVEN CONDITIONS.
163
( 2 )
( 2 )
(5)
(4)
(3, 1)
P =2S
Q =2r
J [=t(m— 3) supra] ;
R =x(m— 3)
m 2 — m+8ft+a( — 3);
ft 2 +8m— ft+a( — 3);
— 3 mn + 9ft+a(fti— 3).
(4)
( 2 )
(4)
N[=/ supra] ;
0[=/c supra].
128. I make the following calculations, serving to express in terms of Zeuthen’s
Capitals, the terms in { } contained in the twelve equations respectively.
N = — 3 m + a
— 3m+a (first equation).
2 J= — 6m 2 + 18m +a(2m6)
+ R= — 3 mn + 9 n +«( m— 3)
— 6m 2 — 3mft+18m+9ft+a(3m— 9) (second equation).
6K=
3ft 2 +24m—
•3 ft+«(
9)
+ L= 
3ft 2
— 9ft j ct(n •
3)
+ 3N=
 9m
+«(
3)
120=
■6ft+a(
2)
15m
+a(ft
7) (th
ird equation).
E=
\m 2 n
—
2m 2 
•mft
+ 4ft 2 + 2m—
16ft qa(
fft+ 6)
+ F =
m 3

4m 2 +
8 mn
j 3m —
24ft+a( —
3m
+ 9)
+2G=
2mft 2 +16m 2 —
2mft
— 8ft 2 — 64m +
■ 8ft + a( —
6m
+24)
+ D = 
fm 3
+
— 18?ft
+ a(im 2 —
fm
+ 6)
+ 3J =

9m 2
+2 7m
+ «(
3m
 9)
+ J' =
— 3ft 2 J
9ft + a(
ft 3)
m 3 
+2mft 2 +
— 7ft 2 — 50m—
■23ft + a(m 2 —
 + 33)
(fourth equation).
2 A
MDCCCLXVIII.
164
PROFESSOR CAYLEY’S SECOND MEMOIR ON THE
Q=tt 2 + 8m— n— 3a
w 2 + 8m— n— 3a (fifth equation).
3G= 3wm 2 +24m 2 — 3mn— 12w 2 — 96m+12w+a( —9m +36)
+ I = — 3 mn 2 +9mw+12w 2 — 36w + u(mn — 3 m— 4w+ 12)
+ 4J = —12m 2 +36m +a( 4m —12)
+2J' =  6w 2 +18rc+a( 2» 6)
12m 2 +6mw— 6w 2 — 60m— 6n{a(mn— 8m— 2w+30)
(sixth equation).
B =^m 3 n —2m 3 — fm 2 w + 4mw 2 + 10m 2 — 14mw— 16w 2 — 8m + 64% co
+ 4C = mV + 8m 3  m 2 n—9mn 2 — 72m 2 + 9mw + 20w 2 +160m— 20w w
+ 4D= —6m 3 +42m 2 — 72m o)
+ D'= — fw 3 + 2 1 ^ 2 — 18w (»
m ! «+mV — f m 2 w — 5m+ 2 — fra 3 — 20 m 2 — 5mra+ 2 2 % 2 + 80m+26ra (*>
o) +«( — fmra + 6m+6ra— 24)
( 2 ) j«(— 3m 2 +27m —60)
(3) +a( 2m 2 —14m +24)
(■t) +a( "fra 2 — fra+ 6)
(5) + a( — m 2 — fmra+fra 2 +19m+^ra— 54) (seventh equation).
— ^N=m — fa
+fO= 4ra+fa
m— 4ra + a (eighth equation).
(2, 3)=4m4ra 6+3«
+2(4, 1)= 2m+2»12
— 2m— 2ra— 18+ 3a (used infra )
— 2P = — 2m 2 + 2m— 16ra+a( 6)
— Q= — ra 2 — 8m+ w+a( 3)
— 2R= 6mra — 18ra+a(— 2m + 6)
+ J' = — 3 ra 2 + 9ra+a( n — 3)
*{(2, 3) + 2(4, 1)}= 6mra + 6ra 2 + 54ra+a(2mllra18) + 3a 2
— 2m 2 + 12mra+2ra 2 — 6m+30ra+a( — 4m— lOra— 6) + 3a*
(ninth equation).
CURVES WHICH SATISFY GIVEN CONDITIONS. 165
3L = — 9 m 2 + 27m+«( 3m — 9)
+2M= 9 m 2 + 3 m+«(6m l)+« 2
— 2N= + 6m +a( — 2)
— O = 3m+os( — 1)
6m+33M+a( — 3 n — 13) + a 2 (tenth equation).
<% 2, 1)=
— 18mM— 18 m 2 —
162m
(i)
D =
W
—
fm 2 + 6m
(2)
— 2E
— m?n
+
4m 2 + ImM— 8m 2 — 4m +
32m
(3)
— 2F =
2m 3
+
8m 2 — 16mM — 6m+
£
oo
(4)
 G =
— mn 3
—
8m 2 + lmM+ 4 m 2 f 32m—
4 n
(5)
+H =
—\m?n
+ \rrm— 4m 2
(6)
+11 =
—bmn 2
+15mM+20M 2 —
60m
(7)
W =
+ 10m 2 —30m
(8)
+2D' =
■3w 3
+21m 2 
36m
(9)
 J' =
3m 2 —
9m
(10)
— fm 3 — fm 2 M— 6mM 2 — Sm^^tm 2 — ^mM+18M 2 — 2m— 191 m
0)
+ “(
— 3mM 
3m 2 +
6m+
51m + 54)+a 2 (m+M
15)
(2)
+ *(“
§m 2
+
m
 2)
(3)
+a(
3m 12)
(4)
+ «(
6m
18)
(5)
+a(
3m
12)
(6)
+ a (

m+
"IpM )+« 2 (
 i)
(7)
+ «(
%mn

5m—
^m 420)
(8) ’
fa(

J#m+10)
(9)
+«(
1m 2
—
7m +12)
(10)
+«(

M + 3)
(11)
+ «(
— fmM
2m 2 + '
4pM +55) + a 2 (m+M
¥)
(eleventh equation).
2 a 2
166
PEOFESSOE CAYLEY’S SECOND MEMOIE ON THE
*( 2 , 1 , 1 , 1 )=
1
M * 1
T
3 mV— 3 mw 3 — ^ 4 +^m 2 w4 24wm 2 + i ^m
‘ — 37mw—
37rc 2
1
Oshf>.
>
II
1
■^m 4
4  fm 3 — %m 2 n
+ fm 2 4  fwm—
¥^ 2
— fm 3 w
+ ^m 3 4  ^m 2 n — ^rnn 2
— ^m 2 4^m^4
^ 2
II
Q
•^w
1
—
\m?n 2 — fm 3 4  \rrfn\ %mn 2
4  2 4m 2 — Smn—
2D =
3m 3
21 m 2
 D' =
fm
1
nf ^ 2
— jm 4 — ^m a n — 2 £m 2 n 2 — 3wm 3 — \it + 4m 3 + ~^m 2 n + ym» 2 + 9n 3 — ^m 2 — 1 4mn — f %z 2
+ 2 2 5 w + a(fm 3 + m 2 w + mn 2 + — fm 2 — fnrn + 2 n 2 + n — 7 5 ) 4 a 2 ( — f m — f n 4   2 )
fm4~ w + a( w 2 — m+ 8w— l) + a 2 ( — f)
(3) + ^f%+a(
<4) — Ypm4 ^^4a(
(■>) 4  36m 4«(
( 6 ) 4 18w4«(
4fmw
—10m— 10^440)
— 9m 420)
4  7m 12)
+
6 )
— i 3 % 4  1 f lM +“(i m 3 + TO!!}l 4 » 2 + 6 fi3 f w& 2 — ww4f^ 2 — fw— 34)4a 2 (— fm— f 413)
(twelfth equation)
129. We have consequently, by means of the results just obtained,
Supp. (5) = a(5)
+N
Supp. (4", 1)
= *(4,1)
42J 4R . . .
Supp. (3, 2)
= *(3, 2)
46K4L43N+20
Supp. (3, 1, 1)
= *(3,1,1)
4D+E4F+2G43J4J' . . .
Supp. (2, 3)
= *(2,3)
4Q •
Supp. (2, 2, 1)
= *( 2 , 2 , 1 )
+ 3G4_I4_4J + 2J'
Supp. ( 2 , 1, 1, 1) = <2, 1,1,1)
4B44C+4D4D' ....... (seventh equation)
CURVES WHICH SATISFY GIVEN CONDITIONS.
167
Supp. (1,4) = *(1,4) +(m— )(4N+20)
iN+fO .
Supp. (1, 1, 3) = *(I, 1, 3)+*(2, 3)+*.2(4, 1)
+(m2)(2P+2Q+5J+4E)
— 2P— Q— 2E+J' . . .
Supp. (T, 2, 2) = *(T, 2, 2)
+ (m— 2)(9K + 3L M+2N + 0)
+ 3L + 2M2N0
(eighth equation)
(ninth equation)
(tenth equation)
Supp. (1, 1, 1, 2) = *( 1, 1, 1, 2) +*(2, 2, 1)
+(m— )(3E+3F+6G+2D+ H+2I+ 5J)
2E2F GiD+iH+I^J+2D'J'
(eleventh equation)
Observe that
G— 2E'=0, G'— 2E=0,
and
3G+I+8J=3G' + r + 8J',
relations which may be used to modify the form of the last preceding result.
Supp. (1, 1, 1, 1, 1)= *(1, 1, 1, 1, l)+*(2, 1, 1, 1)
+ (mf)(A+2B+4C+3D)
— fA— B— fC— D' .... (twelfth equation)
130. We may in these equations introduce on the righthand sides in place of a
symbol such as the symbol : for example, in the fifth equation, writing
(2, 3)=(2^T, 3) + [(2, 3)(2 ST, 3)],
and therefore also
*(2, 3) =*(2*1, 3)+*[(2, 3)— (2*1, 3)],
the second term *[(2, 3)— (2*1, 3)] can be expressed in terms of Zeuthen’s Capitals.
The remark applies to all the twelve equations ; only as regards the first four of them,
inasmuch as (5*1)=0, . . (3*1, 1, 1)=0, it is the whole original terms *(5) . . *(3, 1, 1)
which are thus expressible by means of Zeuthen’s Capitals. By the assistance of the
formulse (First Memoir, Nos. 69 and 73) we readily obtain
Referring to
*(5) — z=0 (first equation)
*(4, 1) =*(m+w— 6)
=R+J' (second equation)
168
PROFESS OE CAYLEY’S SECOND MEMOIR ON THE
*(3,2) =*( — 9 + a)=*(3(% — 3)+* — 1 + 1) Referringto
= 3L+2M+0 (third equation)
*(3, 1, 1) =*(> 2 +2mw+i% 2 ^i%+27f«)
=H+2I+D'+J' (fourth equation)
viz. * _1 H =fm 2 — fm +4% —fa
* _1 . 21 = 2wm — 6m — 8w+24
* , .D' = — §w+ 6
* _1 J' = n— 3
fm 2 + 2?7m + — L 2 m — 1 £ j n + 2 7 — f a.
*(2, 3) =*(2*1, 3) (fifth equation)
*( 2, 2, 1) =*(2*1, 2, l)+*(w3)
=*(2*1, 2, 1) + J' (sixth equation)
*(2, 1, 1, 1)=*(2*I, 1, 1, l)+*.i(»3)(w4)
=*(2*1, 1, 1, 1)+D' (seventh equation)
*( 1, 4) =*(1*1, 4)+*
=*(1*1, 4)+0 (eighth equation)
*(T, 1, 3)+ *(2, 3) +*2(4, 1)
=*(1*T, 1, 3)+*(M, 3) +*(»3)
+*(2*1, 3)
+ *(2m + 2w— 6)
=*(1*1, 1, 3)+2*(2*l, 3) +2R + 3J' .... (ninth equation)
*(1,2,2) =*(1*1, 2, 2)+*{3(w— 3)+*— 1}
=*(1*1, 2, 2)+3L+2M) (tenth equation)
*( 1 , 1 , 1 , 2 )+*( 2 , 2 , 1 )
=*(1*1, 1, 1, 2)+*(2*l, 2, l)+*{f(w— 3)(%— 4)+^+2 3 m— 4}
+*(2il, 2, l)+*( w _3)
=z(lzi, 1 , 1 , 2)+2*(2*l, 1 , 2)
+D'+H + 2I+J' (eleventh equation)
CURVES WHICH SATISFY GIVEN CONDITIONS.
169
*(1, 1, 1, 1, l) + *(2, 1, 1, 1) Referring to
=*(LH, 1, 1, 1, 1)+ *(>1, 1, 1, 1)
+ *(2*1, 1, 1, l)+z.}(n3)(n4)
=*(1*1, 1, 1 , 1 , 1)+2*(2*1, 1 , 1 , 1) + D' .... (twelfth equation)
131. Hence, substituting in the expressions of the several Supplements, we have
Supp. (5)
= 0
+N
(first equation)
Supp. (4, 1)
= R+J'
+2J+R
(second equation
Supp. (3, 2)
= 3L + 2M+0
+6K+L+3N+20
(third equation)
Supp. (3, 1, 1)
= H+2I+D'+J'
+E+F + 2G+D + 3J+J' ....
(fourth equation)
Supp. (2, 3)
= *(2*I, 3)
+Q
(fifth equation)
Supp. (2, 2, 1)
= *(2*1, 1, 1)+J'
+ 3G+I+4J+2J'
(sixth equation)
Supp. (2, 1, 1, 1)
= *(2*1, 1, 1, 1,)+D'
+B + 4C + 4D+D'
(seventh equation)
Supp. (I, 4)
= *(LH, 4) + 0
+(m)(4N+0)
iN+0
(eighth equation)
Supp. (I, 1, 3)
= *(la, 1, 3) + 2*(2*l, 3)+2R + 3J'
+(m2)(2P+2Q+5J+4R)
— 2P— Q2R+J' . . .
(ninth equation).
Supp. (1, 2, 2)
= *(LH, 2, 2)+3L+2M
+ (m2)(9K+3L + M+2N+0)
+ 3L+2M2N0 .
(tenth equation).
170
PROFESSOR CAYLEY’S SECOND MEMOIR ON THE
Supp. (I, 1, 1, 2) = *(I*1, 1, 1, 2)+2*(2Z, 1, 2)
+H+2I4D'+ J'
+ (m— )(3E+3F + 6G+2D+ H+2I+ 5J)
2E2F GiD+H+I^J + 2D'J'.
(eleventh equation)
Supp. (1, 1, 1, 1, 1)= *(1*1, 1, 1, 1, 1)+2*(2*1, 1, 1, 1)+D'
+(mf)(A+2B+4C+3D)
— A — B— fC— 2D— D'. (twelfth equation)
132. Hence finally, merely collecting the terms, we have the following expressions
of the Supplements in the twelve equations respectively.
Supp. (5) =N+0 (first equation)
Supp. (4,1) =2J+2R+J' (second equation)
Supp. (3, 2) =6Kj4Lf2M+3N30 (third equation)
Supp. (3, 1, 1) =DfE+Fj2GjH + 2I+3J+D'+2J 7 (fourth equation)
Supp. (2, 3) =*(2*1, 3) + Q (fifth equation)
Supp. (2, 2, 1) =*(2*I, 2, 1) + 3G+I+4J+3J' (sixth equation)
Supp. (2, 1, 1, 1) =(*2*1, 1, 1, 1)+B44C+4D + 2D' (seventh equation)
Supp. (1,4) =*(1*1,4) +(4m— 7)N+(2m— 1)0 (eighth equation)
Supp. (1, 1, 3) =*(1*1, 1, 3) +2*(2*1, 3)
+(2m— 6)P+(2m— 5)Q+(5m— 10)J+(4m— 8)B+4J' . (ninth equation)
Supp. (T, 2, 2) =*(1*1, 2, 2)
+(9m— 18)K+3mL+(m+2)M + (2m— 6)N+(m— 3)0 . (tenth equation)
Supp. (I, 1, 1, 2) =*(L d, 1, 1, 2)+2*(2a, 1, 2)
+(2 m 5)D+(3m 9)E+ (3m 9)F+(6m 15)G
4(m— l)H(2m— l)I+(5m— 15)J + 3D' .... (eleventh equation)
Supp. (T, 1, 1, 1, 1 )=*(ffl, 1, 1, 1, 1)+2*(2*1, 1, 1, 1)
+ (m— 4)A+(2m— 7)B+(4m— 12)C+(3m— 10)D, . (twelfth equation)
where I recall the remark, ante, No. 126, that in each equation the Capitals belong to
the system obtained by diminishing the barred number by unity and removing the
bar; (4) for the first equation, (3, 1) for the second, and so on.
CURVES WHICH SATISFY GIVEN CONDITIONS.
171
133. These are, I think, the true theoretical forms of the Supplements, viz. (attending
to the signification of the Capitals) the expressions actually exhibit how the Supplement
arises, whether from proper conics passing through or touching at a cusp, or from point
pairs (coincident linepairs) or linepairs (including of course in these terms linepair
points). Thus, for instance, Supp. (5) = N } O. Referring to the explanations, First Me
moir, Nos. 41 to 47, N(=/) is the number of the linepairpoints described as “inflexion
tangent terminated each way at inflexion,” and 0(=%) the number of the linepairpoints
described as “ cuspidal tangent terminated each way at cusp,” or in what is here the
appropriate point of view, we have as a coincident linepair each inflexion tangent
and each cuspidal tangent. Reverting to the generation of the first equation, when the
point P is a point in general of the given curve, the curve 0 is the conic (5), having
with the curve 5 intersections at P, and besides meeting it in the 2m— 5 points P'. When
the point P is at an inflexion, the curve © becomes the coincident linepair formed by
the tangent taken twice, the number of intersections at P is therefore =6, and the
inflexion is therefore (specially) a united point. Similarly, when the point P is at a
cusp, the curve 0 becomes the coincident linepair formed by the tangent taken twice,
the number of intersections at P is therefore =6, and the cusp is thus (specially) a united
point: we have thus the total number of special united points agreeing with the
foregoing a posteriori result, Supp. (5)=N+0.
134. Or to take another example ; for the fifth equation we have
Supp. (2, 3)=a(2^I, 3) + Q;
Q( = 2r) is the number of the linepairpoints described as “double tangent terminated
each way at point of contact,” or, in the point of view appropriate for the present purpose,
we have each double tangent as a coincident linepair in respect to the one of its points
of contact, and also as a coincident linepair in respect to the other of its points of
contact. Reverting to the generation of the equation, when the point P is a point in
general on the given curve, the curve 0 is the system of conics (2, 3) touching the curve
at P, and having besides with it a contact of the third order ; since for each conic the
number of intersections at P is =2, the total number of intersections at P is =2(2, 3),
and the remaining (2m — 2)(2, 3) intersections are the points P'. Suppose that the point
P is taken at the point of contact of a double tangent ; of the (2, 3) conics, 1 (I assume
this is so) becomes the coincident linepair formed by the double tangent taken twice,
and gives therefore 4 intersections at P, the remaining (2, 3)— 1 conics are proper
conics, giving therefore 2(2, 3) — 2 intersections at P, or the total number of intersections
at P is 2(2, 3)j2 intersections; or there is a gain of 2 intersections. As remarked
(No. 96), this does not of necessity imply that the point in question is to be considered as
being (specially) 2 united points ; I do not know how to decide a priori whether it is to
be regarded as being 2 united points or as 1 united point, but it is in fact to be regarded
as being (specially) only 1 united point ; and as the points in question are the 2r points
of contact of the double tangents, we have thus the number 2 r of special united points.
mdccclxviii. 2 B
172
ON THE CUKVES WHICH SATISFY GTVEN CONDITIONS.
Again, when the point P is at a cnsp, all the (2, 3) conics remain proper conics
((2*1, 3)=(2, 3), First Memoir, No. 73), but each of these ( qua conic touching the cuspidal
tangent) has with the given curve at the cusp not 2 but 3 intersections, so that the total
number of intersections at P is 3(2*1, 3), =3 (2, 3), and there is a gain of (2 3) =(2*1, 3)
intersections. Each cusp counts (specially) as (2*1, 3) united points, and together the
cusps count as *(2*1, 3) united points; we have thus the total number *(2*1, 3)+2r of
special united points, agreeing with the expression Supp. (2, 3)=*(2*1, 3)+Q.
135. As appears from the preceding example, or generally from the remark, ante ,
No. 96, 1 have not at present any a priori method of determining the proper numerical
multipliers of the Capitals contained in the expressions of the several Supplements.
I will only further remark, that the reason is obvious why (while in the first seven
equations the multipliers are mere numbers) in the eighth and following equations
the multipliers are linear functions of m; in fact in these last equations the barred
symbol is 1, that is, when P is a point in general on the given curve, each of the conics
which make up the curve © has with the given curve not a contact of any order, but an
ordinary intersection at P. Imagine a position of P for which one of these conics be
comes a coincident linepair ; this regarded as a single line has with the given curve
( m—a ) ordinary intersections (a a number, =4 at most, depending on the contacts
which the line may have with the curve); for each of the m—u points, taken as a posi
tion of P, one of the conics which make up the curve © becomes the coincident line
pair, and there are in respect of this conic two intersections at P instead of one inter
section only. We have thus in respect of the particular coincident linepair a group of
(m — a) special united points, viz. these are the m — a ordinary intersections of the coin
cident linepair regarded as a single line with the given curve, and we thus understand
in a general way how it is that the order m of the given curve enters into the expres
sions of the multipliers of the several Capitals in the last five equations. The object of
the present Memoir was, however, the a, posteriori derivation of the expressions {ante..
No. 132) of the twelve Supplements; and having accomplished this, but being unable to
discuss the results with any degree of completeness, I abstain from a further discussion
of them.
C 173 ]
V I. Addition to Memoir on the Resultant of a System of two Equations.
By Professor Cayley, F.R.S.
Keceived August 6, — Eead November 21, 1867.
The elimination tables in the Memoir on the Resultant of a System of two Equations
(Phil. Trans. 1857, pp. 703715), relate to equations of the form (a, b . . ^) m =0,
without numerical coefficients ; but it is, I think, desirable to give the corresponding tables
for equations in the form ( a , 6, . .fx, y) m = 0 with numerical coefficients, which is the
standard form in quantics. The transformation can of course be effected without diffi
culty, and the results are as here given. It is easy to see a priori that the sum of the
numerical coefficients in each table ought to vanish ; these sums do in fact vanish, and
we have thus a verification as well of the tables of the present Addition as of the tables
of the original memoir, by means whereof the present tables were calculated.
Table (2, 2).
Resultant of
(a, b, cjoc, yf ,
(P> P r X^ yf'
MDCCCLXVIII.
174
PROFESS OK CAYLEY’S ADDITION TO MEMOIR
Table (4, 2).
Resultant of
(a, b, c, d, e\x, y)\
{p, q, rjcc , y)\
Table (3, 3)*.
Resultant of
(a, b, c, djx, yf
(p, q, r , sjx, yf .
* N.B. In the corresponding
table of the memoir, there is an
error in the signs of the last
two terms ; they should be
ON THE RESULTANT OF A SYSTEM OF TWO EQUATIONS.
175
Table (4, 3).
Resultant of
(a, b, c, d, ejx, y)\
(P> r, sjx, y)\
2 c 2
176
PROFESSOR CAYLEY’S ADDITION TO MEMOIR
Table (4, 4).
Resultant of
(a, b, c, d, ejx, y)\
(jo, q , r , s, #) 4 .
ON THE KESULTANT OF A SYSTEM OF TWO EQUATIONS.
177
178
ON THE EESULTANT OE A SYSTEM OF TWO EQUATIONS.
179
180
ON THE RESULTANT OF A SYSTEM OF TWO EQUATIONS.
[ 181 ]
VII. On the Theory of Local Probability, applied to Straight Lines drawn at random in
a plane ; the methods used being also extended to the proof of certain new Theorems
in the Integral Calculus. By Morgan W. Crofton, B.A., of the Royal Military
Academy, Woolwich ; late Professor of Natural Philosophy in the Queen's University,
Ireland. Communicated by J. J. Sylvester, F.B.S.
Received February 5, — Read February 27, 1868.
1 . The new Theory of Local or Geometrical Probability, so far as it is known, seems to
present, in a remarkable degree, the same distinguishing features which characterize
those portions of the general Theory of Probability which we owe to the great philo
sophers of the past generation. The rigorous precision, as well as the extreme beauty
of the methods and results, the extent of the demands made on our mathematical
resources, even by cases apparently of the simplest kind, the subtlety and delicacy of
the reasoning, which seem peculiar to that wonderful application of modern analysis —
ce calcul delicat, as it has been aptly described by Laplace — reappear, under new forms,
in this, its latest development. The first trace which we can discover of the Theory of
Local Probability seems to be the celebrated problem of Buff on, the great naturalist* —
a given rod being placed at random on a space ruled with equidistant parallel lines, to
find the chance of its crossing one of the lines. Although the subject was noticed so
early, and though Buffon’s and one or two similar questions have been considered by
Laplace, no real attention seems to have been bestowed upon it till within the last few
years, when this new field of research has been entered upon by several English mathe
maticians, among whom the names of Sylvester and WooLHOUSEf are particularly
* The mathematical ability evinced by Buffon may well excite surprise ; that one whose life was devoted
to other branches of science should have had the sagacity to discern the true mathematical principles involved
in a question of so entirely novel a character, and to reduce them correctly to calculation by means of the inte
gral calculus, thereby opening up a new region of inquiry to his successors, must move us to admiration for a
mind so rarely gifted.
f Many remarkable propositions on the subject, by these eminent mathematicians, have appeared in the
mathematical columns of the £ Educational Times’ and other periodicals. A very important principle has been
introduced by Professor Sylvestee, which may be termed decomposition of probabilities. For instance, he has
shown that the probability of a group of three points, taken at random within a given triangle, fulfilling a given
intrinsic condition ( i . e. one depending solely on the internal relations of the points among each other), may be
expressed as a linear function of two simpler probabilities ; viz. that of the same condition being fulfilled
(1) when one of the points is fixed at a vertex of the triangle, and a second restricted to the opposite side ;
(2) when all three points are restricted, one to each side of the triangle. The order of the integrations required
MDCCCLXVII1. 2 D
182
ME. M. W. CEOFTON ON THE THEOEY OF LOCAL PEOB ABILITY.
distinguished. It is true that in a few cases differences of opinion have arisen as to the
principles, and discordant results have been arrived at, as in the now celebrated three
jpoint problem, by Mr. Woolhouse, and the fourpoint problem of Professor Sylvester;
but all feel that this arises, not from any inherent ambiguity in the subject matter, but
from the weakness of the instrument employed; our undisciplined conceptions of a
novel subject requiring to be repeatedly and patiently reviewed, tested, and corrected by
the light of experience and comparison, before they are purged from all latent error.
The object of the present paper is, principally, the application of the Theory of Pro
bability to straight lines drawn at random in a plane ; a branch of the subject which
has not yet been investigated. It will be necessary to begin by some remarks on the
general principles of Local Probability. Some portion of what follows I have already
given elsewhere* *.
2. The expression “ at random ” has in common language a very clear and definite
meaning; one which cannot be better conveyed than by Mr. Wilsox’s expression
“ according to no law.” It is thus of very wide application, being often used in cases
altogether beyond the province of mathematical measurement or calculation.
In Mathematical Probability, which consists essentially in arithmetical calculation,
when we speak of a thing of any kind taken at random, there is always a direct refer
ence to the assemblage of things to which it belongs and from which it is taken, at
random, — which here comes to the same thing as saying that any one is as likely to be
taken as any other. When we have a clear conception of what the assemblage is, from
which we take, and not till then, we can proceed to sum up the favourable cases.
In many problems on probability there is no difficulty in forming a clear conception
of the total number of cases. Thus if balls are drawn from an urn, the number of cases
is the number of balls, or of certain combinations of them ; and if the number of balls
be supposed infinite, no obscurity arises from this. But there are several classes of
questions in which the totality of cases is not merely infinite, but of an inconceivable
nature. Thus if we try to imagine how to determine completely by experiment the
probability of a hemisphere thrown into the air falling on its base, we may suppose an
infinite number of persons to make one trial each ; afterwards we may suppose each
person to make two, three, or an infinite number of trials ; again, we may suppose for
every trial that has taken place an infinite number of others, varying, for instance, in
the substance, size, &c. of the body employed; and so on. We can thus continually
suppose variations of the experiment, each variation giving a new infinity of cases
Now problems of this nature are treated by means of the following principle : —
In any question of probability regarding an infinite number of cases, all equally pro
is thus reduced by three. The same method applies to any polygon, and also to the points taken in space within
a tetrahedron. It is to be hoped that Professor Sylvester will give these remarkable results to the public in a
detailed form : a general account of them was given to the British Association at Birmingham in 1865.
* Educational Times, May 1867.
ME. M. W. CEOFTON ON THE THEOET OF LOCAL PEOBABILITY.
183
bable, the result will be unaltered if we take, instead of these cases, any lesser infinity
of cases, chosen at random from among them*.
3. The case of a point or straight line taken at random in a plane or in space is a
problem of the above description. Thus, if a point be taken at random in a plane, the
total number of cases is of an inconceivable nature, inasmuch as a plane cannot b e filled
with mathematical points, any infinitesimal element of the plane containing an unlimited
number of points. We see, however, by means of the above principle, that we may
consider the assemblage we are dealing with, as an infinity of points all taken at random
in the plane.
Let us examine the nature of this assemblage. As the points continue to be scattered
at random over the plane, their density tends to become uniform. It is evident, in fact,
that a random point is as likely to be in any element dS of the surface, as in any equal
element dS' ; and therefore by continuing to multiply points, the number in dS will be
equal (or subequal, to use a term of Professor De Morgan’s) to that in dS'. Of course,
though the density tends to become uniform, the disposition of the points does not tend
to become symmetrical; those within any element dS will be dispersed in the most
irregular manner over that element f. However, it is important to remark that, for all
purposes of calculation , the ultimate disposition may be supposed symmetrical ; for as
the position of any point is determined by that of the element dS, within which it falls,
it matters not what arbitrary arrangement we assume for the points within the element.
* This proposition, of which, in a somewhat different form, a mathematical demonstration is given by
Laplace (Theorie Analytique des Probability, chap. 3), may be regarded as almost axiomatic. Thus, suppose
an urn to contain an infinite number of black and white balls, in the proportion of 2 to 3 ; if any lesser infinite
number of balls be drawn from it, the black ones among them will be to the white as 2 to 3. For, imagine all
the halls ranged in a row ACB, the black from A to C, the white from C to B ; if we now select an infinite
number at random from among them, it appears selfevident that, if the line be divided into five equal parts, the
numbers of balls taken from each part will be the same, or rather, will tend to equality on being increased inde
finitely. Hence the black balls selected will be to the white as AC to CB, or as 2 to 3. When the numbers are
large, but not infinite, this principle is approximately true, and forms, as is well known, the basis of most of the
practical applications of Probability. Thus the chance of an infant living to the age of twenty is as truly found
from, say, 1,000,000 of observed cases, as it would be from the total number.
In its strict mathematical form, the proposition may be thus stated : — In any unlimited number of cases,
divided into favourable and unfavourable, if p be the ratio of the favourable to the whole number of cases, and
if we select any infinite number of cases at random from among them, the probability is infinitely small, that the
same ratio, as determined from the selected cases, shall differ from p by a finite quantity.
+ Order thus results from disorder, the uniform density of the aggregate being unaffected by the disorder and
irregularity of arrangement of its ultimate constituents ; much as a nebula of uniform brightness is related to
the stars which compose it. This remarkable law is to be traced, under one form or another, in most of the
applications of the Theory of Probability.
“Elle merite l’attention des philosophes, en faisant voir comment la regularity finit par s’etablir dans les
choses meme qui nous paraissent entierement livrees 'au hasard.” — Laplace.
A familiar illustration of the tendency to uniform density in the random points may be derived by observing
the drops of rain on a pavement at the commencement of a shower : as the drops multiply, it will be evident to
the eye that their density tends more and more to uniformity.
2 D 2
184
ME. M. W. CROFTON ON THE THEORY OE LOCAL PROBABILITY.
Hence we may, if we please, assume that, when a point is taken at random in a plane,
those from which it is taken are an infinite number symmetrically disposed over the plane.
Likewise, points taken at random in a line may be supposed equidistant. And if
random values be taken for any quantity , they may be supposed to form an arithmetical
series, with an infinitesimal difference.
Let us now consider the case of a straight line drawn at random in an infinite
plane : the assemblage from which we select it is, as before, an infinity of lines drawn
at random in the plane. What is the nature of this aggregate 1 First, since any direction
is as likely as any other, as many of the lines are parallel to any given direction as to
any other. Consider one of these systems of parallels ; let them be cut by any infinite
perpendicular. As this infinite system of parallels is drawn at random, they are as
thickly disposed along any part of the perpendicular as along any other; the inter
sections being in fact random points on the perpendicular. Hence it is easily seen that,
for all purposes of calculation, the assemblage of lines may be thus conceived. Divide
the angular space round any point into a number of equal angles 18, and for every direc
tion let the plane be ruled with an infinity of equidistant parallel lines, the common
infinitesimal distance being the same for every set of parallels. Or we may suppose one
such system of parallels drawn, and then turned through an angle 18, then through
another equal angle, and so on, till they have returned to their former direction.
If we take any fixed axes in the plane, a random line may be represented by the
equation
x cos 8\y sin 8=p,
where p and 8 are constants taken at random.
There is no difficulty in extending now our conceptions to points, straight lines, and
planes, taken at random in space.
4. We may take any plane area as the measure of the number of random points
within it : in the case of random lines, I proceed to prove the following important prin
ciple : —
The measure of the number of random lines which meet a given closed convex plane
boundary , is the length of the boundary.
Draw any system of parallels meeting the boundary, their common infinitesimal
distance being Ip. If we take this distance as wnity, the number of
these parallels is AB, a line cutting them at right angles. Let
AB=S, and let 8 be its inclination to any fixed direction in the
plane; conceive now a consecutive system of parallels inclined to
the former at an angle od, then a third, and so on, till the parallels
return to the direction in the figure ; then the total number of lines
will be
4 ^ »
or, if O be any fixed pole inside the boundary, and OV =p, the perpendicular on the
Fig. 1.
AWT
a
MR. M. W. CROFTON ON THE THEORY OF LOCAL PROBABILITY.
185
tangent to the boundary, $ its inclination to a fixed axis, the measure of the number of
lines * is
Now the integral extended through four right angles gives the whole length of
the boundary , whatever be its nature, provided it be convexf.
Hence if L be the length of the boundary,
N=L.
This result may be obtained also as follows. It may be shown very simply by the
above principles that the measure of the number of random lines which meet any finite
straight line of length «, is ‘la (it may indeed be assumed as selfevident that the number
is proportional to a). Conceiving now the boundary L as consisting of straight elements,
the number of lines meeting any element ds , is Ids ; so that the whole number which
meet the boundary would be 2L ; but as each line cuts the boundary in two points, we
should thus count each line twice over ; hence the true number is L.
Hence if L be the length of any convex boundary, and l that of another, lying wholly
inside the former, the probability that a line drawn at random across L shall also inter
sect l, is
v l
*=L‘
It is important to observe that the measure of the number of lines which meet any
nonconvex boundary is the length of a string drawn tightly round it ; as is obvious on
consideration. The same is true for a boundary which is not closed.
5. Let there be any two boundaries external to each other : let X be the length of
an endless band passing round both, and crossing between them, and Y the length of
another endless band also enveloping both, but not crossing ; then the measure of the
number of random lines which meet both boundaries is X— Y.
It will be easily found from the principles explained above, that the number required
will be the integral j fdb (referred to O as pole), taken for the
lefthand curve from the position RR' of its tangent, to the
position PO ; then for the righthand one from the position P'O
of its tangent, to the position S'S ; then for the lefthand one,
from SS' to QO ; then for the righthand one, from Q'O to R'R.
Now the values of these integrals are, drawing the perpendiculars
OV, OW to RR', SS',
* It 'will be well to remember that this measure of the number of lines, N, means the actual number multi
plied by the constant factor SO. Our notation is thus simplified, and no confusion need arise from sometimes
saying “ the number of lines,” for shortness, instead of “ the measure of the number of lines.” As SO remains
constant throughout our investigations, henceforth we will denote it by S.
t As L=j^ £dd, we see that the mean breadth of any convex area is equal to the diameter of a circle whose
circumference equals the length of the boundary. By breadth is meant the distance between two parallel tangents,
whose direction is supposed to alter by uniform increments.
186
ME. M. W. CEOETON ON THE THEOEY OE LOCAL PEOBABILITY.
1. the mixed line KPO — RV,
2. „ „ S'P'O  S'W,
3. „ „ SQO SW,
4. „ „ R'Q'O  R'V,
and the sum of these is evidently equal to X— Y.
I will add a different proof of this proposition, deduced from art. 4, as it is interesting
to see our results verified.
For shortness, I will use the symbol N(S) for “ the number of random lines meeting
the space S and N(S, S') for the number meeting both S and S'.
The number of lines meeting both boundaries is evidently identical with the number
meeting both the mixtilinear figures OPHQ, OP'PI'Q'. These two figures together form
the mixtilinear reentrant figure HPP'H'Q'Q, and by art. 4, N(HPP'H'Q'Q)=Y.
Now N (OPHQ) + N (OP'R'Q') = N (HPP'H'Q'Q) + N (OPHQ, OP'H'Q'). But OPHQ,
OP'H'Q' being convex figures, the number of lines meeting each is represented by its
length; therefore
X=Y+N(HPQ, H'P'Q').
The probability that a line drawn at random across a given convex boundary of length
L shall also meet a given external boundary is therefore
XY
P=IT‘
6. If two convex boundaries L, L' intersect each other, in two or more points, it may
be proved in a similar manner that the number of random lines which meet both is
represented by L+L' — Y, where Y is the length of an endless band passing round both.
Hence the probability that a line which meets L shall also meet L', is
L + L'Y
P=— L
7. It may easily be proved that the measure of the number of random lines which pass
between two given convex boundaries is
N = PP' + QQ' — arc PQarc P'Q',
where PP', QQ' are the two common tangents which cross each other.
Thus the number of random lines which pass between the two branches of an hyper
bola is represented by A, the difference between the whole length of the hyperbola and
that of its asymptotes. This difference, as is known, is given by the definite integral
A=4 a( \/l—e 2 sin 2 0.cW,
Jo
where sina=—
e
8. Two lines are drawn at random across a given convex area : to find the probability
of their intersection lying within the area.
ME. M. W. CROETON ON THE THEORY OE LOCAL PROBABILITY.
187
Let AB be the internal portion of any random line crossing the area : the number of
its intersections with all the random lines in the area is the number of
2AB
those lines which meet it. Now this number is — y (art. 4); hence
the number of intersections of the system of parallels to AB with all
the random lines in the area, is twice the sum of the lengths of all
these parallel chords divided by <$. But this sum is the area of the
figure (we have taken the common distance Ip of the chords as unity).
2fl
Let O be the area, L the length of the boundary. As, then, y is the number of inter
sections for any system of parallels, and the number of those systems is the total number
of intersections is But we have thus counted each intersection twice ; so that the
real number of intersections which fall inside the area O is
Hence the required probability is
/L
since the whole number of intersections is \ ^y
Thus it is an even chance that two random chords of a circle intersect within the
circle ; for any other figure the chance is less than
If an infinity of lines are drawn at random in an infinite plane, the density of their
intersections (L e. the measure of the number* of intersections in any given space,
divided by the space) is uniform, and equal to t.
9. If an infinity of random lines meet a given area, the density
of their intersections , at any external point P, is
q — 6— sin 6,
where S is the apparent angular magnitude of the area from that
point.
Conceive an infinitely small circle, or other figure (whose dimensions, however, infi
nitely exceed hp), at P, and let us calculate the number of the said
intersections which fall inside this circle. Let the figure represent
this circle, magnified as it were ; QV, RW being the tangents PA, PB.
Draw one of the random lines CD, which meet both the circle and
the area Q, the actual number of intersections which lie on CD will
be iN(Q, CD), which is found from art. 5 to be
y (2CD— CH — Cl),
Eig. 4.
Eig. 3.
* We take for this measure the actual number multiplied by §6' 2 , or S : (see note, art. 4).
188
MR, M. W. CROFTON ON THE THEORY OF LOCAL PROBABILITY.
CD,
^2 — cos a— cos ($ — .
Hence the actual number of intersections on all the chords parallel to CD is
^ (area of circle) ^2 — cos a , — cos (0 — a)^.
Therefore the measure* of the whole number of intersections lying within the circle is
(area)J (2— cos a— cos (0— a)jdu=(area)(0— sin 0),
which proves the theorem.
10. The number of the intersections external to the given area is, then, measured by
the integral
jj(0— sin 0)dS
extended over the whole plane outside O ; dS being the element of the area. Now
the number of internal intersections is tO (art. 8), and the sum of both is We
obtain thus, in a singular manner, the following remarkable theorem in Definite Inte
grals : —
If 6 be the angle between the tangents drawn from any external point (x, y) to any given
convex boundary , of length L, enclosing an area 0, then
JJ (0 — sin 0)dxdy—\ L 2 — sr!2,
the integration extending over the whole space outside D.
It does not seem easy to deduce this integral, in its generality, by any other method.
It may be verified by direct integration for the cases of a circle, and of a finite straight
line. It forms a striking example of what will doubtless be found, as the study of
Local Probability advances, to be one of its most remarkable applications, viz. the
evaluation of Definite Integrals. All who have studied the subject must have remarked
the variety of ways in which almost every problem may be considered ; now it often
happens that a question in which we are bafiled by the difficulties of the integration,
when we attempt it in a particular way, may be solved with comparative ease by other
considerations : we can then return to the integrals which we were unable to solve, and
assign their values. I proceed to give some further applications of the above theory to
Integration.
11. Given any infinite straight line outside a given convex boundary of length L, let
dx be any element of this line ; a, (3 the inclinations of dx to the two tangents drawn
from it to the boundary, then
f (cos 05+ cos j3)^’=L.
* We take for this measure the actual number multiplied by 50 2 , or £ 2 (see note, art. 4).
MR. M. W CEO FT ON ON THE THEORY OF LOCAL PROBABILITY.
189
It is easy to see from art. 5 that the number of random lines
cutting L, which also meet dx, is dx ( cos a+ cos (3) ; now the
sum of all such elements gives the number of lines cutting both
L and the given infinite straight line ; that is, L (art. 4). This
integral may be otherwise verified.
If the boundary L be enclosed within any outer convex boundary, let ds be the differ
ential of the length of the latter, a, j 8 the inclinations of ds to the tangents from it to L,
then we find in the same manner,
J(cos a+ cos/3)2c. The same will appear by means of elliptic coordinates*.
15. I will mention the following integral here, as, though strictly not derived from
the theory which forms the subject of this paper, the principle used in obtaining it is,
as in the cases which precede, the calculation of the number of intersections lying on a
given space, of a given reticulation of straight lines.
Given a closed convex boundary without salient points ; if we draw an infinity of
tangents to it, each making an infinitesimal angle (£) with the preceding, and consider
the intersections of all these tangents with each other, it will not be difficult to show (as
in art. 9) that the number of intersections lying on any element dS will be
8 2
sin 6
TV
d S,
* The general integral above admits also of being established by means of a certain generalization of elliptic
coordinates, which defines the position of a point by the sum and difference of two strings, each of which is
attached to a fixed point on a given oval curve ; they are then wrapped round the curve in opposite directions,
and leave it as two tangents, meeting and terminating at the proposed point.
MR. M. W. CROFTON ON THE THEORY OF LOCAL PROBABILITY.
193
where T, T' are the tangents from d& to the boundary, and Q their mutual inclination .
Now the whole number of tangents is and that of intersections
therefore that
2tt 2
We infer
sin 9
TV
dS = 2v\
the integral extending over the whole external surface.
If the integral extend over the annulus between the given boundary and an outer line
along which Q has a constant value (a), then
If the same integral extend over the space between the given boundary and two fixed
tangents , including an angle a, its value will be J(t— a) 2 . If it extend over the infinite
angle formed by those tangents produced, its value will be c5 2 .
If the given boundary contain salient points, then for every such point, where the
bounding line changes direction abruptly through an angle A, a number of the tangents,
equal to meet at that point; hence a number °f intersections coincide there,
and consequently we must subtract ^A 2 from each of the above integrals. Hence if
there are any number of salient points A A' A", &c. in the boundary, the first integral
becomes
J’^(?S=2 ! r ! iSA s ,
and likewise for the second.
Thus for a regular polygon of ( n ) sides, the value is
If instead of drawing tangents to the given boundary at uniform angular intervals,
we draw a system of tangents whose points of contact are distant from each other by a
common infinitesimal interval, we shall find that the density of the intersections in this
case varies as
Iff sin 0,
where §§' are the radii of curvature of the boundary at the points of contact of TT' : this
gives us the integral
ff ^7 sin $dS= ^L 2 ,
L being the whole perimeter of the boundary, the integral extending over the whole
plane.
Many analytical definite integrals may be deduced by expressing the general theorems
now given, in the language of different systems of coordinates, for various particular
194
ME. M. W. CEOFTON ON THE THEOEY OE LOCAL PEOBABILITY.
cases. Thus the first theorem in this article, applied to the ellipse, gives
i i^y 2 + b^x 2
f
{ [x 2 + y 2 + c 2 ) 2 — 4 c 2 # 2 } V cPy 2 + W'x 2 — a L b‘ i
dxdy—Tr 2 ,
• l TO*" r U*'
the equation of limits being >1.
16. Let there be a closed convex area &>, length of boundary l, en
closed within another of length L ; let 3 be the apparent magnitude
of a at any external point ; by considering two systems of random
lines, one crossing the boundary L, and the other l, and examining
the law of the density of the intersections of the former with the
latter, we arrive at the theorem : — if we put for shortness
a — sin a=u a ,
S( w e+,
27 rr — 2g>.
If dS be an element of the surface at P, the sum of the favourable cases will be
F=jJ(27rr2 ? )^S=2jVr f ).2T^ f ;
.. F=(7T— 1)2^.
But the whole number of cases is 27rr X vr 2 ; hence the required chance is
2
P = 1 si
Fig. 12.
I will give another solution of this problem : — Let AB be a position of the
line; take MN=r, then all the favourable positions of the random
point are within the segment ETIF ; the number of favourable points
is therefore
r 2 {%—$ f sin
+$t
If a random point and a random straight line be taken within any convex boundary
of length L, the chance that the line shall pass within a distance D of the point, D being
small, is approximately,
2ttD
2. If three lines are drawn at random across a given circle, to determine the proba
bility that their three intersections shall lie within the circle.
Let AB be one of the random lines. The total number of favourable
triads of random lines, each triad of which includes AB, is the same as
the number of intersections, which fall within the circle , of all random
lines which cross AB. For every such intersection which lies within
the circle, gives a pair of lines meeting AB, forming a triad whose
intersections all lie within the circle. Now if 0 be the angle which
AB subtends at any internal point P, the number of these intersec
tions will be measured by (art. 9)
N =jj(0— sin 0)dS,
extended over the whole circle.
To integrate this, conceive the circle divided into an infinite number of elementary
crescents, by segments of circles on AB ; let O be the centre of the segment APB, q its
radius; then the area of the segment APB is, putting AB=2«,
segment =(tt — 0)f\aq cos 6, orasg>sin0=«
ME. M. W. CEOFTON ON THE THEOET OF LOCAL PEOBABILITT.
197
Differentiating this for 0, we obtain for the area of the crescent between APB and the
consecutive arc on AB,
crescent = (l +( 7 r—0) cot fjd0.
Hence the number of intersections above AB will be
N=2a’£ (1+Ord) cot 6)de ;
N f" ( Odd 9 cos 9d9 9 2 cos 9 , dQ cos QdQ 9 cos 9 , 1
’* 2a 2 J a sin 2 9 ' ^ sin 3 9 sin 3 9 sin 9 ^ sin 2 9 sin 2 9 j
All these are elementary integrals, and give (reducing the indeterminate forms by the
usual methods)
N 3 a 2 7T — a 7t net
2a 2 2 sin 2 a sina ' 2 C0 a ^~2sin 2 a
To find the number of intersections below AB, change a into ir— a; this gives for the
whole number of favourable triads (including AB),
N=2a 2 (3— ^ ;
y sin a 1 sin 2 a )
or if c be the radius of the given circle, «=c sin a ;
N=2c 2 (3 sin 2 a— , fig. 12) of the restiform
body, covered by arciform fibres, as shown at A, fig. 13, which represents a lateral view
of the medulla oblongata.
(8) Having thus briefly described these elementary structures of the medulla oblon
gata as they are seen in transverse sections, I will now exhibit them in longitudinal dis
sections, and employ, as far as possible, the same letters to indicate the same parts.
The posterior surface of the medulla oblongata, especially in the neighbourhood of
the fourth ventricle, presents a greater diversity of appearance, amongst different indi
ME. J. L. CLAEKE ON THE INTIMATE STEUCTUEE OF THE BEAIN.
267
viduals in Man, than in mammalia of the same species ; but the general arrangement of
the parts is nevertheless the same in all. Fig. 11, Plate IX. represents the posterior
aspect of the medulla oblongata of a healthy middleaged man who died in consequence
of an accident: a is the cut surface of the lower end of the medulla, near the level of
the points of the anterior pyramids. Adjoining the median sulcus, on each side, is the
posterior pyramid (5), which, as it ascends to the point of the calamus scriptorius, ex
pands into a thick bulbous mass ( b '), and then diverges as a flattened band. External
to the pyramid is the restiform body (c), which also enlarges as it ascends to the same
level. Between this and the posterior edge of the anterolateral column ( d ) is a super
ficial tract of grey substance (e), consisting of the expansion of the caput cornu, or
dilated extremity of the posterior horn, and known as the grey tubercle of Bolando.
(9) At the point of the calamus scriptorius, on each side of the median line, is an
oval or pyriform mass of ganglionic substance (g), of a bluish or pearly hue, constituting
the superficial part of the nucleus of the vagus nerve, seen in section at g , fig. 9, Plate
VIII. ; and the small tract ( t ') between this and the median fissure is the upper part of
the nucleus of the hypoglossal nerve, seen in section in fig. 10, Plate VIII. ; its lower
end being covered in by the spinalaccessory nucleus and the posterior pyramids ( t , b\
figs. 6, 7 & 8, Plate VIII.). In the angle between the outer side of this vagal nucleus
( g , fig. 11, Plate IX.) and the upper divergent end of the posterior pyramid (5'), is the
commencement of another and larger mass of grey substance (i), which is seen in section
in fig. 10, Plate VIII., and forms the posterior nucleus of the auditory nerve, covered by
epithelium.
(10) If the posterior pyramid b V {on the left side ) be carefully dissected from the
restiform body (c), from below upwards, and he thrown forward, as shown in fig. 12,
Plate IX., the exposed surface of the medulla will present the appearances delineated
at the lower half of this figure ; t is the downward continuation of the vagal nucleus or
tract, constituting the nucleus of the spinalaccessory nerve, and previously covered in
by the bulb of the posterior pyramid (b 1 ), as shown at t, fig. 7, Plate VIII. in a trans
verse section. The oval mass g, marked off by the dotted line, is the inner and posterior
portion of the vagal nucleus exposed at the point of the calamus scriptorius by the
divergence of the posterior pyramid ; l is its inner and more anterior portion, covered ,
like the spinalaccessory nucleus t, lower down, by the posterior pyramid, as seen in
transverse sections, figs. 8 & 9, Plate VIII. Its upper point ( m ) forms the principal
nucleus of the glossopharyngeal nerve. Along the outer and anterior part of this grey
tract is a slender, longitudinal white column (n), which it lodges, as it were, in a groove
(see n, fig. 9, Plate VIII.), and which tapers to a point as it descends obliquely inward
along the base of the posterior pyramid to the mesial line (see n, fig. 12). In its course
upward it ascends along the inner edge of the pyramid, and joins those fibres of the
latter which pass into the anterior or outer auditory nucleus (see n, fig. 42, Plate XII.,
and fig. 58, Plate XIV.). On the outer side of this slender white column is a some
what fusiform mass of grey substance, o (fig. 12, and oo', fig. 24, Plate X.), imbedded
268 ME. J. L. CLAEKE ON THE INTIMATE STETJCTUEE OE THE ERA TN.
in the inner side of the restiform body, and exposed by the removal of 'the posterior
pyramid. From the upper end of this fusiform grey mass a thin but broad layer of
fibres, mixed with some grey substance (p), radiates upward and outward on the resti
form body (see also p, fig. 24).
Fig. 14, Plate IX. is another careful dissection of some of these parts. The pos
terior pyramid (b) has been separated on the left side of the medulla from the inner side
of the posterior column ('), and the grey tubercle of Rolando, e
(caput cornu posterioris), are still further enlarged. The decussating fibres
from b o, the postpyramidal and restiform nuclei, instead of entering the
anterior pyramids y y, now proceed to the lower ends of the olivary bodies
(W, W). The spinalaccessory nucleus (t) and the hypoglossal nucleus (t')
are still further developed, r is the spinalaccessory nerve.
Fig. 7. A similar section of the medulla nearer the point of the calamus scriptorius.
The inner and outer restiform nuclei ( o and d) are much enlarged, and the
white substance (c) of the restiform body is overlain by the posterior pyramid
( b ) and its contained nucleus ( V\ also increased in size. The hypoglossal nerve
(w) is seen to enter its nucleus (l 1 ), which, like the spinalaccessory nucleus (t) is
further developed. The olivary bodies (W) increase in size ; s is the antero
lateral nucleus..
Fig. 8 is a similar section just below the calamus scriptorius. The posterior median
fissure has just begun to open into the fourth ventricle. The grey nuclei of
the posterior pyramids and restiform bodies nearly coalesce, the dotted line,
b 11 (on the right side), defining the limits of each. The olivary bodies are
much increased in size.
Fig. 9. A similar section through the fourth ventricle, a little above the calamus scrip
torius. The grey nuclei of the posterior pyramids and of the restiform bodies
on each side have entirely coalesced. The spinalaccessory nucleus has now
become the vagal nucleus (g), und is more distinctly separated from the
hypoglossal nucleus (t 1 ). At U is seen a small group of cells forming the
lowest end of the nucleus of the motor root of the trigeminus.
Fig. 10. Similar section of the medulla on a level with the upper roots of the vagus
nerve. The hypoglossal nucleus (t 1 ) is brought to the surface of the ventricle,
while the vagal nucleus (g) is partly sunk between it and the rudiment of the
inner auditory nucleus (i) which is developed out .of the posterior part of the
vagal nucleus (g) and the inner part of the posterior pyramid (b, fig. 9). The
convolutions of the olivary bodies are more numerous than in fig. 9.
PLATE IX.
Fig. 11. Posterior view of human medulla oblongata: — a, its lower end; bb\ the poste
rior pyramid of right side ; c, right restiform body ; d, lateral column ; e, grey
tubercle of Rolando, or caput cornu posterioris ; g , pyriform nucleus of vagus
ME. J. L. CLARKE ON THE INTIMATE STRUCTURE OE THE BRAIN. 325
nerve; /, column at side of median furrow constituting the hypoglossal
nucleus.
Fig. 13. Side view of the same: — W, olivary body; y, anterior pyramid; A, arciform
fibres.
Figs. 15 to 21 represent transverse sections of the medulla oblongata of the Monkey;
fig. 15, section just below the point of the anterior pyramids. The same
letters signify the same parts as in figs. 1 to 10 of the human medulla, Plate
VIII.
PLATE X.
Fig. 22. Transverse section of the posterior part of the right lateral half of the medulla
oblongata of the Monkey at the level of the highest roots of the vagus nerve.
Fig. 23. Similar section of the whole right lateral half of the medulla: — i, the inner
auditory nucleus ; d o, the outer auditory nucleus, pierced by numerous lon
gitudinal bundles of fibres (represented by the dark masses) ; t", nucleus of
the lower roots of glossopharyngeal nerve ( g ).
Fig. 24. Longitudinal and horizontal section of the left half of the posterior part of the
human medulla oblongata : — b b', white and grey substance of the posterior
pyramid; o, inner grey substance or nucleus of restiform body; d, its outer
nucleus ; g, vagal nucleus ; i, inner auditory nucleus ; d o, outer auditory
nucleus, formed by the fusion of the restiform and postpyramidal nuclei.
Fig. 25. Transverse section of the spinalaccessory and hypoglossal nuclei of the left side,
at the level of the lower end of the olivary body : — E, bottom of the posterior
median fissure ; H, group of cells constituting the anterior portion of the spinal
accessory nucleus ; r, the spinalaccessory nerve ; the remainder of the figure
behind this group is the posterior division of the spinalaccessory nucleus ;
B, columns of cells and longitudinal fibres, with transverse fibres decussating
across the middle line with those of opposite side ; below these are decussating
fibres from the anterior division (H) of the nucleus ; h, central canal lined by
columnar epithelium; J, hypoglossal nucleus; J", fanshaped set of fibres
converging to decussate across the raphe (F) ; K', small column of cells and
longitudinal fibres, ascending and increasing as it ascends, between the hypo
glossal and spinalaccessory nuclei to the “ fasciculus teres ; ” x, roots of
hypoglossal nerve.
Fig. 26. Nervecells from the outer restiform nucleus of the Monkey; magnified 220
diameters.
Fig. 27. Obliquelongitudinal section of the medulla oblongata, carried along the line
L t, fig. 5, and extending as high as fig. 8 : — n n' is the slender column of the
vagus and spinalaccessory nerves ; 1 1 is vagus and spinalaccessory nucleus ;
the dark masses at n" represent the cutends of the transverse bundles of fibres
326 ME. J. L. CLAEKE ON THE INTIMATE STEITCTITEE OF THE BEAIN.
shown at n", fig. 5, and coming from the postpyramidal and restiform nuclei
to decussate into the anterior pyramids.
Fig. 28. Left lateral half of the grey substance of the medulla, a little below fig. 5, and
where the lower roots ( r ) of the spinalaccessory nerve begin : — t , lower end of
the special nucleus of the upper roots of the nerve ; t', lower end of the special
nucleus of the upper roots of the hypoglossal nerve, at the sidefront of the
canal ; x, lower hypoglossal roots arising from a group of cells in the remains
of the anterior cornu, f ; o, restiform nucleus ; b', postpyramidal nucleus. On
the left of t f are seen the last traces of the curved bundles of fibres which
the anterior pyramids in their course downward send into the posterior grey
substance (see n", fig. 5).
Fig. 29. Transverse section of the grey substance of the lowest end of the medulla
oblongata at the point of the anterior pyramids : — e", posterior roots of first
cervical nerves extending forward into the anterolateral grey substance, to
the part where the lowest fibres (/') of the anterior pyramids are connected
with it. The same part is also connected with the lower roots of the spinal
accessory nerve ( r ), which is seen to traverse the lateral grey substance and
reach the anterior cornu.
Fig. 30. Longitudinal section of part of the human medulla: — t", glossopharyngeal
nucleus ; n n, slender longitudinal column ; g, roots of the glossopharyngeal
nerve running partly longitudinally along this column ; i, the inner auditory
nucleus.
Fig. 31. Longitudinal and horizontal section along the floor of  the fourth ventricle, at
the dotted line, fig. 32, through the hypoglossal nucleus (J), the vagus nucleus
(H, H'), and the slender longitudinal column (n). From the large multipolar
cells of the hypoglossal nucleus (J, J", fig. 31), numerous processes extend
outward, between longitudinal bundles of fibres, into the inner and outer
portions (H, H') of the vagus nucleus, n is the slender column of longitudinal
fibres, continuous with some of the transverse fibres of the vagus nucleus.
PLATE XI.
Fig. 33. Transverse section of the medulla oblongata of the Codfish, at the point of the
calamus scriptorius : — 1, 1, the anterior portion of the medulla; 2, 2, cutends
of large bundles of fibres at its surface ; 3, group of large multipolar cells in
its centre ; 4, cutend of column of longitudinal fibres in front of the canal, 6 *
5, large group of multipolar cells at the side and front of the canal ; 7, fibres
radiating from it to the surface (on left side) ; 8, lower root of vagus
nerve arising from it, and joining the other root, 9, which arises from the
inner and posterior part of the medulla (10, 10) at the side and behind the
canal. The group of cells (5) occupies the position of the hypoglossal nucleus
ME. J. L. CLAEKE ON THE INTIMATE STEUCTUEE OF THE BEAIN.
327
in mammalia. 11, restiform body; 12, grey substance within it; 13, large
band of fibres arising from it, and crossing obliquely to the anterior part of
the opposite side of the medulla, decussating with its fellow ; 14, trunk of
the vagus nerve.
Fig. 34. Group of multipolar cells giving origin to the anterior root (8) of the left vagus
nerve, as shown in preceding figure ; — 6, left wall of the canal lined with epi
thelium; 13, 13, decussating band of fibres; magnified 100 diameters.
Fig. 35 represents a transverse section of the right anterior pyramid (y) and olivary body
(W) of the Cat. The olivary body consists of three separate portions.
Fig. 36 represents a transverse section of the right lateral half of the medulla of one of
the higher Apes. The structure is peculiar. The highly developed and
convoluted olivary body (W) is imbedded in the anterior pyramid, and a
separate column ( d ) is on its outer side.
Fig. 37. Transverse section of the right anterior pyramid and olivary body of the Orang
Outang. The olivary body largely developed, and its lamina thrown into
numerous folds.
Fig. 38. Transverse section of the floor of the fourth ventricle of the left side, from the
human medulla, on a level with the upper roots of the vagus nerve : — J, upper
part of column of large multipolar cells constituting the hypoglossal nucleus.
Some of the cells on its outer side are elongated outward, and send their pro
cesses in the direction of the vagus nucleus (t, H) and of the nerve (y). It is
immediately overlain by the column K', which consists of smaller cells and
numerous longitudinal fibres, and which we saw lower down, much reduced in
size, in fig. 25. This column is in its turn overlain by the vagus nucleus,
between which and the hypoglossal nucleus (J), it is almost entirely imbedded.
If we suppose the lower part of the central canal ( h ) in fig. 25 to open into
the floor of the fourth ventricle, and the parts on its left to be thrown aside
outward, the three nuclei (J, K', H, fig. 25) will assume the position which
they occupy in fig. 38.
Fig. 39. Similar section a little higher up, on a level with the roots of the glossopha
ryngeal nerve. The hypoglossal nucleus has almost wholly disappeared, a few
small cells only (J, J') remaining with a few nerveroots. T, dark oval group
of cells, increased in size from fig. 38 ; K', in the fasciculus teres , much
increased in size; i, inner nucleus of auditory nerve; t, glossopharyngeal
nucleus with its clubshaped group of cells. It is overlain by the junction of
the auditory nucleus (i) with the fasciculus teres ; U, lower end of column of
large multipolar cells constituting the nucleus of the motor root of trigeminus
nerve. It is connected, by a plexiform system of fibres (U'), with both the
fasciculus teres and the glossopharyngeal nucleus, especially the latter ; n,
slender longitudinal column of fibres.
Fig. 40. Similar section still higher up. The inner auditory nucleus (i) has become
mdccclxviii. 2 z
328 ME. J. L. CLAEKE ON THE INTIMATE STEUCTUEE OE THE BEAIN.
continuous with the fasciculus teres (K'), and increased in size: — Q, Q',
cut ends of oblique linece transversal ; T, the two oval groups of nuclei much
increased in size.
Fig. 41. One lateral half of the human pons Yarolii and medulla seen from its under
side : — Y, cut middle peduncle ; X'', fibres winding round it from fourth
ventricle ; c, restiform body ; Z", band of fibres winding round it from fourth
ventricle, to side of pons ; X, facial nerve ; Z, auditory nerve ; Z', cut end of
fifth nerve ; w, olivary body ; Y, anterior pyramid ; W W', pons Yarolii.
PLATE XII.
Fig. 42. Transverse section of the posterior portion of the human medulla on left side : —
c, restiform body ; d o, outer auditory nucleus ; i, inner auditory nucleus ;
K', fasciculus teres ; S, a network containing several large multipolar cells,
with intervening longitudinal bundles beneath the fasciculus teres, and in the
place where the hypoglossal nucleus ends ; H, nucleus of glossopharyngeal
nerve ; n, slender columnar of longitudinal fibres, now containing a great
number of nuclei, and connected with glossopharyngeal nerve ( g ) ; U, nucleus
of motor root of trigeminus ; e eee, cut end of descending root of trigeminus
at extremity of caput cornu posterioris, or grey tubercle; P, posterior audi
tory nerve ; QQ', a large stria medullaris winding over the surface of the
ventricle, and passing from behind forward along one side of raphe, E.
Fig. 43. Posterior half of the left side of the human medulla oblongata, just below the
pons Yarolii: — W, upper end of olivary body; e, grey substance of grey
tubercle overlain in front by the dark cut ends of the bundles of the descend
ing root of the trigeminus ; i, inner auditory nucleus ; d o, outer auditory
nucleus ; P P', posterior auditory nerve arising from both these nuclei, and
from radiating fibres (f) of restiform body. A network of fibres proceeding
from the outer auditory nucleus ( d o ) crosses the grey tubercle ( e ) to the
central part of the medulla and to the raphe ; in the higher Apes these
fibres are very numerous and distinct (see fig. 36, Plate XI.).
Fig. 44. The same, a few sections higher up : — Z, clusters of small cells on the inner
side of the ganglionic enlargement of auditory nerve ; Q', Q, cut ends of strise
medullares.
Fig. 45. Side view of the human medulla oblongata, pons, tubercula quadrigemina, and
crus cerebri : — c, restiform body entering cerebellum ; P", anterior auditory
nerve ; one portion of it ascends with restiform body to cerebellum ; C', tri
geminus ; E, superior peduncle of cerebellum ; f, fillet ; q\ testis.
Fig. 46. Transverse section of left lateral half of the medulla oblongata of a child five
years old, at the anterior auditory nerve : — K', fasciculus teres, connected by
numerous plexiform fibres with U, the motor nucleus of trigeminus ; e , grey
MR. J. L. CLARKE ON THE INTIMATE STRUCTURE OF THE BRAIN. 329
tubercle of Rolando, having in front the cut ends of numerous bundles of de
scending root of trigeminus inclosed in a beautiful network of fibres containing
nervecells. The tubercle is connected with the motor nucleus (U) of trige
minus by fibres ; i, upper part of inner auditory nucleus ; d o, outer auditory
nucleus, extending backward to the cerebellum ; P", the anterior auditory
nerve, entering this nucleus at d o, beneath the restiform body. Both the
nucleus, and the root of the nerve P" are intimately connected with the outer
side of the grey tubercle and with the network within it ; p\ radiating fibres of
restiform body in the middle peduncle ; p", p'", arciform fibres from the same
part extending round the olivary body ; J', corpus dentatum cerebelli.
Fig. 47. Dissected portion of human fourth ventricle of left side : — c d, the restiform
body or inferior peduncle of the cerebellum ; d", middle peduncle of cere
bellum ; C", large root of trigeminus ; D', its conical nucleus ; P", anterior
division of auditory nerve; i, concavity left by removal of inner auditory
nucleus beneath which P" enters the outer auditory nucleus. The anterior
and under portion of the trigeminal nucleus D', connected with the point of
the fillet ( f ), gives origin to the small or motor root (G') of the trigeminus,
which is separated from the large root (C"j by a small bundle of the middle
peduncle (A') coming from the pons. The larger root (C") is separated from
the anterior auditory nerve (P") by a larger bundle of the middle peduncle
coming from the pons ; E', superior peduncle rolled forward, exposing a thick
and broad tract of grey substance (D") extending obliquely upward beneath it
from the conical nucleus (D') of the trigeminus ; F, loop of the facial nerve.
Fig. 48. Left lateral half of the medulla with part of the inferior vermiform process of
the cerebellum from the Rabbit : — e, grey tubercle ; d', cut end of descending
root of trigeminus in front of it ; P", anterior root of auditory nerve ; d", cut
ends of longitudinal fibres at root of grey tubercle, in connexion with fibres of
anterior auditory root ; d o, outer auditory nucleus giving origin to anterior
root of auditory nerve ; i , upper part of inner auditory nucleus ; E' E", plexus
of bundles, with intervening cells running through the lower part of superior
peduncle to vermiform process (I' I"); T", longitudinal portion of loop of facial
nerve ; s, anterolateral nucleus.
PLATE XIII.
Fig. 49. Transverse section of left lateral half of medulla oblongata of Cat, at level of
auditory nerve : — P w , under and whiter portion of nerve entering the outer
auditory nucleus (d.o); P', bulbous portion crowded with nervecells; e", de
scending root of trigeminus ; s , anterolateral nucleus connected with its oppo
site fellow of decussating fibres across the raphe.
Fig. 50. Transverse section of the fourth ventricle in Man on left side: — g" g'", super**
2 z 2
330
ME. J. L. CLAEKE ON THE INTIMATE STEUCTUEE OF THE BEAIN.
ficial grey layer covered with columnar epithelium ; h! 1 , median furrow of
ventricle ; Q", principal nucleus of facial nerve ; i, inner auditory nucleus.
Fig. 51. A similar section a little higher up, including the grey tubercle of Rolando: —
g'", superficial grey layer of ventricle ; T", cut end of longitudinal portion of
loop of facial nerve ; Q", principal nucleus of facial ; q", facial nerve ; e e', grey
tubercle ; c' o, remains of outer auditory nucleus ; i, remains of inner auditory
nucleus, sending fibres to Q".
Fig. 52. Transverse section of fourth ventricle of Rabbit on left side.
Fig. 53 shows the course of the root of the facial nerve ( q ") from without inward in the
Sheep.
Fig. 54 shows the same in Man.
Fig. 55 represents a longitudinal section of loop of the facial nerve in the plane of its
course, in Man : magnified 100 diameters.
Fig. 56 shows a transverse section of the fourth ventricle in Man, on the left side
along the root of the facial nerve : — g", superficial grey layer ; g'", group of
cells on its outer part ; above this are the cutends of three bloodvessels in
their sheaths ; T", cutend of longitudinal portion of loop of facial nerve ;
A", median furrow of ventricle ; Q", principal nucleus of facial nerve ; q", facial
nerve; q'", grey streak in its middle, behind which is a bloodvessel and
streaks of longitudinal fibres. The under portion of the nerve is seen entering
the nucleus (Q"); its remaining fibres decussate with those of the superficial
portion ( l ') of the nerve ; the chief portion enter T" and become longitudinal
along the loop ; the rest pass in front of the loop to the raphe beneath the
median furrow ( A ") ; V', the sixth or abducens nerve, breaking up into a net
work of bundles enclosing longitudinal fasciculi, and becoming connected
with the inner part of Q".
PLATE XIV.
Fig. 57. Horizontal and longitudinal section along the left side of floor of the fourth ven
tricle in Man, extending from the upper end of the hypoglossal nucleus (J)
to the lower end of the facial nucleus (Q ") : — do is the outer auditory nucleus ;
e', the base of the grey tubercle or caput cornu posterioris ; Q", the facial
nucleus; T 7 , lower end of the loop of facial nerve, diverging and curving
outward and downward round Q" ; g"\ the superficial grey layer of ventricle ;
A", the median furrow of ventricle overlying the raphe. At the lower end of
the figure, H is the vagus nucleus ; J, the cells of the upper and tapering end
of the hypoglossal nucleus ; J', a longitudinal plexus of fibres running from
the facial nucleus (Q") to the hypoglossal nucleus (J).
Fig. 58. A similar horizontallongitudinal section a little deeper from the surface of
the ventricle. Figs. 59 & 60 are transverse sections of the medulla at the
MR. J. L. CLARKE ON THE INTIMATE STRUCTURE OF THE BRAIN.
331
upper and lower ends of the longitudinal section, after it was removed. In
figs. 58 & 59, the upper lines of section correspond : — F is the raphe immedi
ately below the median furrow of the ventricle ; V, the white column between
the raphe and the entrance of the abducens (V') into the nucleus (Q") ; d is the
base of the grey tubercle ; do, the outer auditory nucleus. The lower line of
fig. 58 corresponds to the upper line of the transverse section, fig. 60, in which
J is the anterior portion of the hypoglossal nucleus; V the white column
between it and the raphe ; O", the white olivary column lying between it and
the vagus nucleus (H); n, the slender longitudinal column of the vagus;
do', the lower end of the outer auditory nucleus, formed by the grey substance
of the restiform body and posterior pyramid. The white column, Y, (fig. 59),
between the raphe (F) and the entrance of the abducens nerve into the nucleus
(Q") is continuous in fig. 60 with the white column (V) between the hypoglossal
nucleus (J) and the raphe.
Fig. 61. Transverse section of part of the left lateral half of the medulla oblongata of
the Rabbit, through the trapezium: — e, grey tubercle; d', descending root
of the trigeminus; g", facial nerve; p',p", arciform fibres running forward
and inward through the trapezium in front of the superior olivary body, s".
Fig. 62. Transverse section through the whole left lateral half of the medulla at the
level of the facial and abducens nerves, from the Orang Outang : — Q", com
mon nucleus of facial and abducens nerves ; T", descending portion of loop of
facial. Transverse fibres of trapezium (p") more concealed by the enlarged
pyramid ( y ).
Fig. 63. A similar section from Man. Here the transverse fibres of the trapezium' (p")
are completely enclosed and concealed by the anterior pyramid and pons
Varolii.
Fig. 64, A. Atrophied nervecells from the olivary bodies, in a case of paralysis ; magni
fied 350 diameters.
Fig. 64, B. Cells from a healthy olivary body; magnified 350 diameters.
Fig. 65. Transverse section through the fourth ventricle on a level with the facial and
abducens nerves, from a child : the letters indicate the same parts as in other
figures.
Fhil. Trans. KDCCCIXm. Plate W
Fig .
Tig. 2.
70.
Ml. Ttms.W)CCC 1XVIK. Plate IX
Jia. 16.
4S67
Phil . Trans . MD CC C.T.XVTTT , Ply# X
J~' I Clarlte. ad. nat . del. 1867 .
J.Basire
Thil. Trans . MD C C CLXyiH . Plate XI .
Tig. 34 .
Fig. 35.
Fig.
Tig. SZ .
J ' K
J~Z.Cla.rlte. ad. not. del. 1S&7
Tig. //! .
X
J. B as ire
Thil. Trans, MD CC ClJ(.VilI . Plate X 1 1
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‘.del. 1661 .
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J.B <
IHl . Trans . MDCCCIXWH. TlaM TSY.
$fflBm>:a&.nat,a*i . 2667
[ 333 ]
X. Notices of some Parts of the Surface of the Moon By John Phillips, M.A.,D.C.L.,
F.B.S., F.G.S., Professor of Geology in the University of Oxford.
Received January 9, — Read January 16, 1868.
My first serious attempts to portray the aspect of the moon were made with the
noblest instrument of modern times, the great telescope of Lord Rosse, in 1852. The
mirror was not in adjustment, so that the axes of the incident and convergent pencils
of light were inclined at a very sensible angle. This being met by a large reduction of
the working area of the mirror, the performance was found to be excellent. I have
never seen some parts of the moon so well as on that occasion. But when I came to
represent what was seen, the difficulty of transferring from the blaze of the picture to
the dimly lighted paper, on a high exposed station, with little power of arranging the
drawingapparatus, was found to be insuperable, and the effect was altogether disheart
ening. It was like setting down things ex memoria , to give the rude general meaning,
not like an accurate and critical copy. I present as a specimen of this memorial a sketch
of the great crater of Gassendi, marked No. 1. (See p. 344.)
I next mounted, in my garden at York, a small but fine telescope of Cooke, only 2*4
inches in the aperture ; and, aware of the nature of the difficulty which beset me at
Birr Castle, I gave it an equatorial mounting, without, however, a clock movement.
The whole was adapted to a large solid stone pillar in the open air. It was not possible
with ffe of the (reduced) light of the Rosse mirror to see so well ; but it was easy to
represent far better what one saw. A conveniently placed board to hold the drawing
paper, a wellarranged light, no necessity of changing position, — I made in this manner
the drawing of Gassendi which is marked No. 2.
My next attempt was made in the same situation with a fine telescope by Cooke, of
6^inch aperture and 11 feet sidereal focus, mounted equatorially, in the old English
mode, and carried by clockwork. With this excellent arrangement I was enabled to
use photography very successfully, and to obtain selenographs 2 inches across in 5 s of
time. The drawing of Gassendi (Plate XV. fig. 1) was made with this instrument (1853).
From these experiments the conclusion was obvious — that for obtaining good drawings
of the moon convenient mounting was actually more important than great optical power ;
and that for such a purpose it was desirable to increase in every way the comfort of the
observer, and furnish him with special arrangements for his own position and the placing
of his drawingboard and light.
Having been called to reside in Oxford (185354), my plan for continual work on
the moon was entirely cut through ; it was impossible to mount a large instrument near
334
PROFESSOR PHILLIPS ON SOME PARTS
my dwelling till (in 1860) the ground was arranged about the museum, so as to give
me the requisite space and security close to the house which had been appointed for
me by the University. I then arranged with Mr. Cooke for a new telescope of 6 inches
aperture, to be protected in a wellplanned observatory, the construction of which was
aided by the Royal Society. I now propose to give a short notice of some of the results
of my work with this instrument, in connexion with remarks on the most advisable
course to be followed by other surveyors of the moon.
In making drawings of ringmountains on different parts of the moon’s disk, the
artist will be much aided by a projection of the mountainborder on the scale intended,
from a few measures, with its proper figure due to the latitude and longitude. Eye
drafts not thus controlled are apt to become absurd, by the heedless substitution of an
ideal circle for a real ellipse. Thus I have seen Gassendi forgetfully represented in
more than one skilful drawing. Even with the advantage of such a projection (of which
I give an example for Gassendi, No. 5) very considerable difficulties occur. One is the
variation of outline caused by the displacement of the boundary of light and shade, first
when the incidence of light varies through different angles of elevation of the sun, and
next when the moon’s position causes her to receive the light at the point observed on
a different lunar azimuth. Even on so great a ring as Copernicus the variation of the
outline as given by different artists is remarkable — hardly any one agreeing with what
is really the most accurate drawing of all, that by P. Secchi ; and that represents, not
a simple ring, but a sevenangled outline. Dates must always be annexed to the figures ;
and as it is rarely possible to complete a good drawing of a large crater, except in two
or three lunations or more, it becomes very essential that a bold free sketch be made of
the moon’s shadows to control the special work.
Strictly speaking, there should be at least three drawings of a ringmountain — in
morning light, at midday, and in evening. It would be better to have five drawings,
one at sunrise and another at sunset being added to the three already named. It will
be found most convenient to make the drawings within two hours of the moon’s meridian
passage.
Shadows thrown from objects on the moon have exactly the same character as those
observable on the earth. They are all margined by the penumbra due to the sun’s dia
metral aspect; this is always traceable except very near to the object; but in conse
quence of the smaller diameter and more rapid curvature of the moon’s surface, the
penumbral space is narrower. At the boundary of light and shade, on a broad grey
level tract, the penumbral space is about nine miles broad, quite undefined, of course,
but perfectly sensible in the general effect, and worthy of special attention while
endeavouring to trace the minute ridges (of gravel 1) or smooth hanks (of sand X), which
make some of these surfaces resemble the postglacial plains of North Germany, or
central Ireland, or the southern parts of the United States, which within a thousand
centuries may have been deserted by the sea.
To the same cause is due the curious and transitory extension of halflights over some
or THE SURFACE OF THE MOON.
335
portions of the interior of craters, while other neighbouring portions have the full light.
The effect is occasionally to produce halftints on particular portions of terraces within
the crater, as in the case of Theophilus, of which I present two drawings, one (Plate
XVI.) showing this peculiarity in the morning light, the other (Plate XV.) not. The
central mountains of that great crater are high enough to throw long shadows ; and these,
as they catch upon other peaks or spread, softening with distance, over the surrounding
plains, present far greater variety of shadowtones than might be expected on a globe
deficient, as the moon really appears to be, of both air and water.
If we suppose the moon’s mass to have been derived from an outer ring of the earth
nebula, and, following the analogy of the planets in reference to the sun, admit its com
position to have been the same as that of the earth, we should have as the originally out
standing lunar atmosphere, only part of that which surrounds, or rather which
at first surrounded, the earth. The surface of the moon being part of that of the
earth, the barometric pressure o,f the atmosphere on the moon would be less than that
on the earth, in the proportion of The height of the lunar atmosphere, taken in
terms of the pressure at the surface of the moon, would be the same as that on the
earth (5 miles +). The refractions would be less than on the earth, in the proportion
of the function of density, but still very sensible, even in rude observations of occupa
tions of stars.
In like manner the water due to the moon’s mass, would have its depth on the moon
diminished to of that on the earth. If we assume the water on the earth spread
over its whole surface to be 1 mile +, that on the moon would be 829 feet, a quantity
not so great as to preclude the possibility of its being accumulated on the surface
opposed to us, especially if we remember that the inequalities of internal earthmove
ment have in fact occasioned the greater part of our oceans to be collected on one half
of the globe, while nearly all the land appears on the other half. In reference to this
subject the opinion of Hansen, that the centre of gravity in the spherical moon is
removed from the centre of figure 33^ miles in the direction from the earth, may be
kept in mind. If it could be admitted, the oceans of water and air might be wholly
collected on the remote half of the satellite. Moreover, it must be remembered that in
the rocks near the surface of our planet both oxygen in oxides, carbonates, &c., and
water in hydrates, are stored up in considerable quantities. We may suppose the
process of volcanic incineration and aqueous absorption to have gone to greater extreme
in the moon than on the earth, and so to have reduced the original atmosphere and the
original oceans to very much smaller amounts. To judge by the absolute freedom from
all change but that of degradation, which is observed on the visible disk of the moon,
and by the perfect and perpetual clearness of its sharply defined mountainbordered
limb, cutting clear against the unbending rays of the stars, it would seem the most
probable opinion that now at least there remains no visible trace of atmosphere or ocean.
The different parts of the moon’s surface reflect light very unequally ; the dark parts
have several degrees of darkness, the light parts several degrees of light. On the same
mdccclxviii. 3 A
336
PROFESSOR PHILLIPS OJSf SOME PARTS
level, as nearly as can be judged, under the same illumination, neighbouring parts are!
not only unequally reflective, but their light seems to be of different tints. Within the
large area of Gassendi, under various angles of illumination, but more conspicuously
when the angle of incidence deviates least from verticality, patches of the surface appear
distinctly marked out by difference of tint, without shadow. It is well known that in
this particular photography has disclosed curious and unexpected differences of the
light, which were not apparent, or not so obvious, to the eye. Reflecting telescopes
seem to be indicated as most suited for direct observation of differences of the kind of
light on the moon.
The surface of the moon is hardly anywhere really smooth, hardly anywhere so smooth
as may be supposed to be now the bed of a broad sea on our globe. By watching care
fully the curved penumbral boundary of light and shade, — as it passes over ridge and
hollow, rift and plain, — broad swells, minute puckerings, and small monticules appear
and disappear in almost every part. In several of the maria, minute annular cupcraters
about half a mile across are frequent ; and on several of the exterior slopes of the
craterrings occur pits, ridges, fissures, and rude craters, something like the sloping sur
faces of Etna. Copernicus is a good example of this common occurrence. It appears
extremely desirable that the details of this magnificent mountain should be carefully
reexamined on the basis of Secchi’s fine drawing, for the purpose, amongst others, of
determining exactly how many of the bosses and ridges bear cuphillocks ; for many
inequalities, which in feeble telescopes have but the indistinct character of being heaped
up, appear distinctly crateriform with superior optical power*.
For the purpose of determining the true form of the summits, and the outline of
the sides of many mountains, the method of observation of the shadows at different angles
of the incident sunlight will be found very useful. Thus, exactly as, in the clear evening
hours, one standing on the summit of the Malverns sees the long shadow sweep over the
vale of the Severn, and distinguishes the forms of the several beacons and intervening
passes, so in the clearer moonlight, the shadows which fall within the craters, and stretch
along the plains, often reveal the presence of angular escarpments and deep fissures as
well as of peaky summits crowning steep walls of rock.
One of the circumstances which is thus often placed in evidence is the exceeding
abruptness of many edges, and the uncommon steepness of many of the slopes. In par
ticular, the interior edges of many ringmountains appear with violently precipitous
cliffs and chasms, more or less parallel to the general outline ; while, more rarely, deep
cross cuts in such situations appear to radiate within the ring, or to traverse its crest.
An instance of the latter kind is found in Gassendi.
The steepness of the mooncrater walls and slopes is much greater in general than
in any, except very rare examples, known among the volcanic regions of the earth.
Popular descriptions are, in this respect, very misleading ; and the wordpaintings which
please so much the gratified ear, lose their reputation for exactness when confronted by
* See “ Comparative Remarks on Gassendi and Copernicus,” Roy. Soc. Proc. for 1856, p. 74.
OF THE SURFACE OF THE MOON.
337
the clinometer. When, on the surface of the earth, we observe incoherent materials
resting at a slope approaching 45°, we frequently remark some peculiarity of the sepa
rable masses which allows of mutual support, or observe that the usual atmospheric
agencies, the ordinary changes of temperature and moisture, have had only a very short
influence on a newly exposed surface. After a very long exposure, even hard and con
tinuous rocks, such as those of Malvern, sink down to a slope of 18°; and it is chiefly
under freshly wasting escarpments of softer strata that we see, as in Mam Tor and
the west front of Ingleborough, anything approaching to the steepness of the moon
crater walls. So in the old volcanic regions of Auvergne and Mont Dor, the Eifel
and even the Phlegrsean fields, which may be the fittest for comparison with the crater
covered tracts of the moon, very steep slopes are almost unknown, except for small
spaces of consolidated rocks, such as in the Puy de Dome, Monte Somma, and the
Drachenfels.
Perhaps we may refer the obviously greater steepness of some of the surfaces on the
moon to the lesser influence of gravity on the satellite than on the earth. If in a given
mass of matter, exposed to various causes by which cohesion
is influenced, we represent by C (vertical) the residue of co j
hesive force, and by G (horizontal) the force of gravity, the J
C I
slope from C to G will be the final surface, and ^ = tangent I
of its inclination to the horizon. On the earth, the force of &
gravity being more than six times as great as on the moon, the tangent of inclination
will be relatively less.
Also, if the degrading influences of all kinds on the moon be taken as very much less
than on the earth, this again will relatively increase the angle of slope ; so that we
appear to have no difficulty in understanding why lunar mountains may be so much
steeper and less worn than the objects most like them on earth. Perhaps the same
general consideration of the greater relative value of heat and other molecular forces,
when compared with gravity, on the moon than on the earth, may account for the much
greater amplitude, and the generally greater elevation of the moon mountains ; for it
is only exceptionally that Chimborazo, Teneriffe, and Elburz ascend on the earth to so
great a height as the loftiest lunar summits*.
On the very crest of a ringmountain it is rare to find a cupcrater; quite common to
find them in the interior, especially towards the middle, and, in several cases, exactly in
the centre. But it happens often that the central mountainmass of a large crater, such
as Gassendi and Theophilus, is of a different structure. In the former a complicated digi
tated mass of elevated land appeared to me for a long time to be entirely devoid of any
small craters ; by continued scrutiny at last I see on one of the masses a distinct depres
sion. The area in Gassendi reminded me of the volcanic region of Auvergne, in which,
* On the general relation of the moon’s surface to volcanic tracts on the earth, see a graphic passage in
Herschel’s £ Introduction to Astronomy,’ p. 229, Edition 1, 1833.
3 A 2
338
PROFESSOR PHILLIPS ON SOME PARTS
with many craterformed mountains, occur also the Puy de Dome, Puy Sarcoui, Puy
Chopin, and others which are heaps of a peculiar trachyte not excavated at the top,
while the others are formed of ashes and lavastreams, and are all crateriform. The
central masses of Theophilus (Plate XV.) are very, lofty and grandly fissured from the
middle outwards, with long excurrent buttresses on one side, and many rival peaks
separating deep hollows, and catching the light on their small apparently not excavated
tops. This is like the upheaved volcanic region of Mont Dor, with its radiating valleys,
wide in the central part, and contracted to gorges toward the outside of the district.
The Yesuvian volcanic system, including the Phlegrsean fields, exhibits, in all respects
but magnitude, remarkable analogy with parts of the moon studded with craters of
various sizes, as those adjoining Mount Maurolycus, engraved for comparison by
Mr. Scrope in his admirable treatise on Volcanos (p. 232). It is probable that many
of the differences which appear on comparing lunar ringmountains may be understood
as the effects of long elapsed time, degrading some craters before others were set up,
and turning regular cones and cavities into confused luminous mounds. It would much
augment our confidence in the possible history of the moon which these differences seem
to indicate, if we could believe it to have ever been under the influence of atmospheric
vicissitudes as well as changes of interior pressure.
That the latter cause has been in great activity at some early period of the moon’s
history is evident, not only by the many sharply cut fissures which range like great
faults in our earthly strata for five, ten, twenty, and sixty miles, but is conspicuously
proved by the great broken ridges of mountains which, under the names of Alps, Apen
nines, and Riphaean chains, make themselves known as axes of upward movement, while
so many of the craters near them speak of local depression. I have not been able to
discover in these great ridges any such marks of successive stratification, or even such
concatenation of the crests, as might suggest symmetrical and anticlinal axes. The
surface is, indeed, as rough and irregularly broken as that of the Alps and Pyrenees,
and marked by as extraordinary transverse rents, of which one, in the Alpine range near
Plato, is a wellknown example. Must we suppose these mountains to have undergone
the same vicissitudes as the mountainchains of our globe — great vertical displacement,
many violent fractures, thousands of ages of rain and rivers, snow and glacial grinding 1
If so, where are the channels of rivers, the long sweeps of the valleys, the deltas, the
sandbanks, the strata caused by such enormous waste 1 If the broad grey tracts were
once seas, as analogy may lead us to expect, and we are looking upon the dried beds,
ought we not to expect some further mark of the former residence of water there than
the long narrow undulations to which attention has already been called as resembling
the escars of Ireland \
Among other curious phenomena probably referable to movement of the solid crust
of the moon, are those long, straight, winding, or angularly bent fissures or cracks which
the Germans call 4 Itillen.’ The distinctness of these cracks for such great lengths as
some of them reach is consistent with extreme narrowness ; very narrow spaces, strongly
OF THE SURFACE OF THE MOON.
339
contrasted in brightness with the neighbouring surfaces, being easily traceable, if conti
nuous, where much broader tracts of a square or circular figure cannot be discerned.
This is familiar to persons who trace by the eye the far extended narrow telegraph wire.
But more than this. The rills are, under favourable lights, seen to be really fissures with
one dark and one enlightened side — deep dark fissures. Bills of this remarkable cha
racter do not appear to be branched ; and in this respect the long rill of Hyginus, marked
on Madler’s map for ninetytwo miles, may be compared with the great North of
England dyke, which extends from Teesdale to near the Peak on the coast of Yorkshire,
seventy miles, without a branch. Contraction, rather than violent movement of the
moon’s crust, seems to suit best the facts observed ; and the same explanation may be
fairly applied to most of the fissures filled with whinstone in the coalfields of Durham
and Northumberland. Still the straight rill near Tebit (called the Kailroad), nearly on
the central meridian of the moon, bears the appearance of dislocation in the shadow
thrown by one side.
The rill which descends from Herodotus is of a different but equally remarkable cha
racter. Parting from that dark crater — full perhaps of augitic compounds, in strange
contrast with its neighbour Aristarchus, which shines as if formed of white trachyte —
the rill takes a winding course with some irregularity of width and appearance, till it
opens into what resembles an old delta, or dried gulf, margined by cliffs, and undulated
by recesses and promontories. The deltalike space is uneven, like the bed of the
German Ocean. Still, there are appearances in this seeming valley and outlet to an
ancient sea, which can hardly be reconciled to the conjecture. The rill has no branches ;
in the middle of its course is an oblong crater which makes a part of the seeming chan
nel ; and thus this falls into the general rule which makes the rills dependent on the
craters, so as to pass commonly, but not always, from one crater to another, and often
to traverse them through the ring and athwart the interior space (Plate XVII.)*.
A very different class of phenomena may also be referred to some change of dimen
sions or some displacement of masses affecting large surfaces of the moon. These are
the lightstreaks which, from Tycho in particular, radiate, like false meridians, or rather
like meridians true to an earlier pole of rotation. Other lightstreaks pass off from
Copernicus and Kepler, and several other mountains, but none are comparable to those
from Tycho. There is this singularity about them ; they are most distinctly visible in
the full moon, and for some days before and after that phase. When the light falls at
a low angle on the part of the moon’s disk where one or more of these rays exist, they
do not appear ; as the sunlight strikes at higher and higher angles they come out bright
and clear, again vanishing as the lunar evening comes on.
By the strictest examination these luminous bands are found to have no projection
above and no depression beloAv the surface, no shadow on either boundary. They can be
traced across what look like seas, and equally traverse the crateriform mountains, con
tinuously ; sometimes branching, often varying in breadth. On a first view they seem
* See additional notices in the Supplement.
340
PROFESSOR PHILLIPS ON SOME PARTS
like a system of radiating fissures, and Mr. Nasmyth very ingeniously imitated their
direction at least by a glass globe filled with water, which was made to crack in a nearly
regular manner from a centre. But their great and unequal width, their entire want
of relief, and the peculiarity of their reflexion of light, seem to point to some other
cause. The reflexion seems to be of the kind called “ metallic perhaps metals or metal
loids may there be covered by a translucent crust, which may reflect light of low
incidence, but transmit light of higher incidence, to be reflected from the surface below.
This is the nearest conjecture which has occurred to me.
The parts of the moon’s surface to which I have devoted most time are the ring
mountains of Gassendi and Theophilus. The sketches here given of Cyrillus and
Catharina (Plate XVI.), are only first though not careless drawings, in which are
omissions, which I hope at some future time to supply. Bather more progress has
been made with Posidonius, Aristarchus, and Herodotus. I have also begun to sketch
the large and noble group which lies to the south of Ptolemseus, and it is my earnest
hope to be able to finish these drawings as well as to complete a good deal of work on
the Bills.
In any further attempts of my own to contribute facts toward the survey of the moon,
now again taken in hand by the British Association, I shall probably select for careful
work some particular features, such as the mountains in the midst of a large crater, the
bosses and cuplike hills on the outward slopes of such a crater, the rents in mountain
ridges, and the low winding banks which appear on the broad grey tracts. But for
those who desire to perform a work of high value, and lay a sure foundation for accu
rate surveys of particular mountains, I would earnestly recommend a strict reexamina
tion of every element in the great picture of Copernicus, for which we are indebted to
the Boman Astronomer.
The descriptions which follow relate principally to Gassendi and Theophilus.
Gassendi (Plate XV.), whose centre is situate nearly in Lat. S. 16° 56' and in Long.
E. 39° 82', has the long diameter of its apparently elliptical boundary inclined to
the equator about 64° 30'. Measured in this direction, the ring has a diameter of
fortyeight miles ; the shorter diameter is about thirtysix. Sunrise happens when the
moon is 10  5 days old, midday at about eighteen, and sunset at twentyfive days. In
consequence of its position on the moon’s disk, shadows usually fall from elevated objects
a little toward the south.
At the instant of sunrise on the central peaks, the whole of the exterior ring is en
lightened, except for a few miles on the south border, where the crests are low and
divided (18th May, 1853, 10 p.m.). As the early morning advances, very broad shadows
from the western border spread over the interior, and bring out in every part a mingled
effect of unequal height and unequal reflective power of the surface. When the moon
is eleven or twelve days old, the interior appears everywhere diversified by lights and
shades, nowhere smooth, but marked by ridges and hollows in various directions. Within
OF THE SURFACE OF THE MOON.
341
the great ring in all the S.W., S., and S.E. portions are broken ridges, not terrace edges,
separated by considerable depths from the great encircling crest. Toward the northern
part there seems a sort of confluence of small ridges from the interior, directed tow r ard
the spoonshaped crater, marked A on Madler’s map. The mouldings of the surface of
this very deep crater are best seen when the moon is a day older.
Directing our attention to the central mountainmass, we perceive it to be divided
into many digitated parts, each diversified by lights and shades, so as to resemble, more
than anything which I remember, dolomitic or trachytic mountains. On their several
peaks I have sought carefully for the concavities which might indicate former eruptions,
but only on one of them, a little removed from the rest, do I feel sure of this occurrence.
Through these central mountains can be drawn a crescent concave to the S.E., and
along it occur four small craters, three of them rising separately in the area, and the one
already mentioned near to the eastern slope of the central mountain.
The ring of Gassendi, everywhere rugged and fissured, is broken through in one place
completely, in another less distinctly, both in the south part of the contour, so as to open
the area to Mare Humorum. In several other places it sinks between points greatly
elevated. The highest point is on the east border (9000 feet according to Madler),
where a sort of lacuna, rather than crater, occurs in the middle of the crest; nearly
opposite to this ridges detach themselves both outwardly and inwardly from the western
border.
The northeastern edge is cut across by three oblique narrow clefts, and the northern
border is deflected by the concurrence of the ring of the smaller crater. The country
round Gassendi is very ridgy and mountainous in the east, where it is pressed up, as it
were, to the great peak and the lacunal crater. This is not the case on the west, which
side is much lower. On the south, in Mare Humorum, are many small craters not yet
mapped. On the north, in the high grounds leading to the old degraded crater of
Letronne, are a few craters apparently of older date than Gassendi.
As the daylight advances, and especially at midday, much of what has been described
disappears from sight ; but Gassendi, in consequence of its ridges usually having some
shadow toward the south, does not lose distinctness in the same proportion as Ptole
mseus and the craters near the centre of the moon.
When the sun is on the meridian of Gassendi, the peculiarities of its surface in
reflecting light appear conspicuous. Within the south border is a broad dusky space,
apparently not much undulated, and resembling the dark parts of the Maria. This space
sometimes appears interrupted by a cross band of comparative brightness. A darkly
tinted space can also be seen in the N.E. No luminous rays stream from Gassendi.
In the afternoon, the appearances described in the morning are discovered, as far as the
main features are concerned, with shadows in the opposite direction; but the views
have not appeared to me so interesting, nor is the terrestrial hour (early morning) for
observation so convenient.
Theophilus, Cyrillus, and Catharina (Plate XVI.), situated south of the equator from
342
PROFESSOR PHILLIPS ON SOME PARTS
10° to 20°, and in west longitude 21° to 28°, form a remarkable triplet of ringmountains
nearly equal in magnitude, and in contact, yet offering singular differences of structure.
They are not of the same date: Theophilus intrudes its continuous outline within the
northern part of the older crater of Cyrillus ; Cyrillus is linked to Catharina by a con
cave isthmus ; Catharina includes the halfpreserved ring of an older and smaller crater.
In the centre of Theophilus is an aggregation of peaks ; in the same part of Cyrillus
is a huge doublehorned mountain, and near it a considerable crater ; neither of these
features occurs in Catharina.
Theophilus, situated in lat. S. 10° to 13° and long. W. 25° to 28°, though nearly cir
cular in figure, shows some diversity in this respect according to the incidence of light,
being sometimes almost hexagonal. Copernicus also exhibits this diversity.
The longest diameter of Theophilus is above fiftyfive miles in length. Sunrise hap
pens when the moon is 5 ‘2 days old, midday when about 12  7, and sunset at twenty
days.
At sunrise the central mountain and all the surrounding ring are brightly illumi
nated, only narrow channels of darkness running into Cyrillus. But the interior of the
crater is then wholly dark, except for a small breadth within and under the eastern crest
(1863, April 24). When the sun’s rays fall at a low angle on the surface of Theo
philus, the shadow thrown from the lofty western crest (15,000 feet high) spreads widely
over the interior of the deep crater, and is undulated on the edge so as to mark the
effect of higher and lower parts of the crest. Between this shadow and the base of the
central mountain the area appears clear like a floor ; but beyond this, toward the north,
east, and south, all is curiously uneven in heapy little ridges, and long partly fissured
surfaces parallel to the ring. I can discern only one distinct crater in this surface, and
that is under the northeastern part of the ring, but there are several smaller pits.
Perhaps larger instruments may show that some of the small heaps and ridges have cups
on their surface.
To the central mountain (Plate XV.) I have given much and frequent attention,
for the purpose of ascertaining the form of its much divided mass, and of discovering
whether it contained any cupformed summits. None were observed among the ten or
more bosses Avhich go to make up the rugged mass, elevated about 5000 feet above the
central area.
When the sun’s rays fall with less obliquity on Theophilus, the part which was
wholly in shade under the western crest assumes a quite different aspect — many bold
ridges and furrows showing themselves distinctly with light falling at 15° elevation.
Some of these appear in the lower part of the figure in Plate XVI., but they really exist
all round the inner and western part of the crater.
The phenomena which appear where Theophilus joins Cyrillus, are extremely curious
and complicated, not in the least like as if one cone of volcanic eruption had intruded
its convex sloping surface within another, but rather as if one great blister had pushed
aside another and then burst, leaving a sort of double folding along the line of junction.
OP THE SURFACE OF THE MOON.
343
Cyrillus varies "greatly in appearance, according to the direction of the incident light.
The interior is largely rugose, as if ridged and furrowed by lateral pressure. In parti
cular the great double central mountain appears in the two drawings here given (Plate
XV. and Plate XVI.) with a difference of outline very remarkable ; as well as a dif
ference of shadows, for which, however, the sun’s position may perhaps account. The
eastern border is remarkably rugged and broken, so as to become indefinite. The crater
is unenclosed both to the south and north. Catharina seems to contain traces of more
than one old crater enveloped and partly obliterated within it. It is incompletely
bordered, or complicated in the boundary on all sides, and open or unenclosed on the
south and north. If we regard Catharina and Cyrillus as twin craters of old date, broken
into by Theophilus, we may admit the space full of old ridges and hollows to which
Cyrillus opens, on the east of Theophilus, as a third and still older, once crateriform,
mass which has lost almost every trace of the large ancient ring. In the same way the
triplet of larger mountains entitled Arzachel, Alphonso, and Ptolemseus, seem to have
been connected, Ptolemseus being the oldest, largest, and least distinct. These moun
tains, indeed, may be regarded as part of a vast crescent of 60° arc, extending from
Barocius and Maurolycus, by Stoffler, Walter, and Regiomontanus, to Purbach, Arza
chel, Alphonso, Ptolemaeus, and Hipparchus. In each of these cases the oldest moun
tains, those most degraded and ruined, are near the equator. These are among the
greatest of all the lunar mountains.
Another arc, more directly meridional (long. W. 60°), includes the very great rings of
Furnerius, Petavius, Yendolinus, and Langrenus, whose diameters are about 5° of lunar
latitude. The more degraded and probably the older of these are those nearest the
equator.
In various parts of the moon occur twin mountains which do not communicate with
one another ; as Aristarchus and Herodotus, one light the other dark ; Hercules and
Atlas ; Autolycus and Aristillus ; Eudoxus and Aristoteles ; Mercator and Campanus.
Indications of this kind of symmetry or continuity of action in the old volcanic force,
appear worthy of special study, in relation to the physical history of the moon.
Explanation of Plates.
PLATE XY.
Fig. 1. Gassendi, as seen in morning light.
Fig. 2. Theophilus and part of Cyrillus, morning light.
Fig. 3. The central mountainmass of Theophilus enlarged.
PLATE XYI.
Group of Theophilus, Cyrillus, and Catharina.
mdccclxviii. 3 B
344
PEOEESSOE PHILLIPS ON SOME PAETS
PLATE XVII.
The twin craters of Aristarchus and Herodotus, with the rill proceeding from the
latter. Several small craters on the side of the rill and about the seeming delta are
omitted. The crater edge of Aristarchus appears to me a double crest, with a narrow
deep chasm between the ridges. It. The rill from Herodotus. E.W. Borders of the
seeming delta, r. Smaller rill.
The memoir was accompanied by several other drawings, which it is the intention of
the author to present to the Royal Society for reference. The following is a List of
these illustrations. Nos. 1, 2, 4, and 5 are referred to in the Memoir, pages 333 and 334.
No. 1. Sketch of Gassendi, taken in 1852, at Birr Castle, with the great telescope of
Lord Rosse (morning).
No. 2. Sketch of Gassendi, taken in 1852, at York, with an achromatic by Cooke,
of 2 ‘4 inches diameter (morning).
No. 4. Sketch of Gassendi, taken in 1862, at Oxford, with an achromatic by Cooke,
of 6 inches diameter (evening).
No. 5. Working Plan of Gassendi, and Scale.
No. 6. Freehand sketches to illustrate the mode of working for general effect.
Oxford, 1864.
No. 10. Posidonius, early morning, 1863. Unfinished.
No. 11. Posidonius, nearer to midday, 1863. Unfinished.
SUPPLEMENT.
Eeceived May 23, 1868.
Since the preceding remarks were presented to the Royal Society, I have had the oppor
tunity of reading the valuable memoir of J. F. J. Schmidt, the Director of the Observa
tory at Athens, entitled “ Uber Rillen auf dem Monde,” 1866, and of comparing the
maps of some parts of the moon which accompany his essay with my sketches. I
rejoice to see how large an increase to the known number of rills has been made by this
experienced observer, aided by the favourable “ clearness of the Athenian climate.”
In my sketch of Aristarchus (Plate XVII.), the principal purpose was to show the
sharpedged double western crest and the internal broad moulding. In the character of
the double craterwall Schmidt’s Map agrees with my drawing, and extends this feature
to the eastern side.
There is less agreement in the shape of the interior surface, on which I have bestowed
much attention, first with the 3foot mirror of Lord Rosse (power 200), next with my long
achromatic at York, and finally at Oxford with my 6inch Cooke. The country lying
OF THE SURFACE OF THE MOON.
345
to the north of Aristarchus has been five times drawn by me, often with difficulty, for
the appearances are very variable, with small differences of light. My results agree in
general with that of the Athenian astronomer, and with an earlier map published by
Mad lee in 1837. I hope to complete a drawing of this tract.
In Plate XVII. Herodotus and the famous rill from it are merely sketched freely, to
show the general course of the rill, and its opening into a seeming delta. The eastern
border of the seeming delta (E) is treated as a rill by Schmidt and Madlee, and so it has
always appeared to me in the part marked r, which turns up to the south so as to be
rudely parallel to the first or main part of the rill from Herodotus, and seems to begin
under a crest of mountain. The smaller notch in this outline (at E) has only been
noted once. The western branch (IP) is not marked as a rill by Schmidt, but may be
traced on his map as a kind of coast by the line of the promontories, and so traced agrees
with my outline. I propose to complete the drawing of this part of the moon in the
next summer.
Plate 2 of Schmidt’s memoir contains mapsketches of rills lying east of Campanus,
which I have also drawn. The appearances in this region differ somewhat from the
map of Madlee, and the rills in particular are very interesting from their number,
general parallelism, partial concurrence, and distinct relation to small and large craters.
In a third plate of the same memoir is a diagram map of Gassendi, and the region to
the south, which may be compared with my drawing,. Plate XV. fig. 1. The agree
ment is quite satisfactory in the general sweep and breaks of the border, the digitated
central mass, the small craters in the area, the complexity of the eastern crest and
slopes.
There is some difference requiring reexamination in regard to the sculpture of the
interior, and the linear inequalities, some of which are marked as distinct rills by
Schmidt. I shall take an early opportunity of making a careful study of the points of
difference.
.
.
J. Phillips .
GA S S £ A/ O /
after Sunrise
1852. . 1853 . 1863
Thil. Iran s .JED CCCTJCVUI . FlcUeJH.
i THEOPh/LUo apart of CEA/LLi/S.
£ 25 Ma.rc.7u 2863. 8 p.m.. Power ZOO j.OO.
Jilbon's Peel. 15 22 . f Su.ns Peel. 15. If 5 . Of a
6. 23 days.
THE OPH/LUS Z& April 1863.
IPou/er 300 ~ 400.
Tty. 3
Aale of Geographical Miles
Enoravcd bit S. Basb'e
NORTH
10UTH
Phil Trans MD CCCLXVHI, Plate XVI
NORTH
J. Phillips, del.
EAST
WEST
SOUTH
Thu rra^mcccmmiria^mi.
WEST
WH/WesW, Lith.
NORTH
J.PMlips.del.
[ 347 ]
XI. Besults of Observations of Atmospheric Electricity at Kew Observatory , and at
King's College , Windsor , Nova Scotia. By Joseph D. Everett, B.C.L., F.B.S.E.
Communicated by Sir William Thomson, F.B.S.
Eeceived October 14, — Read December 5, 1867.
My papers of June 18th, 1863 and January 12, 1865, contained a record of observations
at Windsor, Nova Scotia, from October 1862 to the end of February 1864. From this
latter date they were continued until August 8th of the same year, and 1 have now to
report this concluding series, giving at the same time a summary of results derived from
the whole of my observations.
I have also to report, at the request of Sir William Thomson, the results of two
years’ observations of atmospheric electricity taken at Kew Observatory with his selfre
cording apparatus, and reduced under his direction and under my own more immediate
supervision at the Natural Philosophy Laboratory of the University of Glasgow.
The concluding series of observations at Windsor were taken regularly at the three
principal hours previously adopted, that is to say, about 9 A.M., 2 p.m., and 10 P.M., but
very few were taken at other hours. The station electrometer alone was used, and the
electricity was collected by burningmatch. The glass fibre mounted in the electro
meter July 31, 1863, remained unchanged to the end of the observations, thus giving a
full year’s observations with the same fibre.
All the generalizations noticed in my former papers apply also to the concluding
series, with the single exception that negative electricity was once observed under a clear
sky. This phenomenon was observed at 10 p.m., July 15th, and as its exceptional cha
racter struck me at the time, it was carefully verified. The negative potential was equal
in absolute amount to about half the average positive potential. The temperature was
59°T, wet bulb 2°*7 below dry, barometer 30T0, no wind, sky perfectly clear, except a
little cirrocumulus near the horizon. There was a faint trace of aurora overhead. The
weather had been fine for several days, and continued so for several days after.
The following Table, showing the mean hourly electrical potential in fine weather
for each month in the concluding series, has been computed in the same manner as the
corresponding Tables in my previous papers. The numbers in this Table are in units
of “ Station Electrometer with third fibre.”
MDCCCLXVIII.
348
DR. EVERETT ON ATMOSPHERIC ELECTRICITY.
1
A
pril.
May.
June.
July.
August.
Total
No. of
Mean of
monthly
No.
No.
No.
No.
No.
Gross
of
Mean.
of
Mean.
of
Mean.
of
Mean.
of
Mean.
obs.
mean.
means. ,
obs.
obs.
obs.
obs.
obs.
5 to
7 A.M.
...
1
17*2
1
17*2
172
1 7 to
8
6
239
5
189
9
234
2
494
2
215
24
246
274
8 to
9
12
321
18
263
14
203
14
264
3
28*7
61
262
268
9 to
10
1
288
3
259
2
23*0
2
316
8
270
273
10 to
11
...
1
336
1
336
336
1 11 to
12
12 to
1 P.M.
1 to
2
16
285
21
249
18
22*3
11
233
1
134
67
246
225
2 to
3
4
200
5
285
5
330
10
21*8
3
273
27
255
26*1
3 to
4
1
322
3
192
1
217
5
22*3
244
4 to
5
"2
458
1
130
3
348
294
5 to
6
6 to
7
7 to
8
8 to
9
9 to
10
12
192
14
20*3
11
109
8
144
45
167
162
10 to
11
7
235
10
253
16
175
12
100
4
153
49
17*9
183
11 to
12
1
189
2
21*7
1
14
1
847
5
297
317
In the following Table, which includes the whole of my fineweather observations from
the beginning, all observations, whether actually taken with first, second, or third fibre,
are reduced to units of second fibre, that being the unit adopted in my previous papers.
Although the numbers for the first seventeen months have already been published in
the Proceedings, it seems desirable to reproduce them, in order to show the connexion
of the whole series.
Mean of
observa
tions before
noon.
Mean of
observa
tions from
noon to
6 r.M.
Mean of
observa
tions after
6 P.M.
Mean of
three pre
ceding
columns.
Mean of
observa
tions before
noon.
Mean of
observa
tions from
noon to
6 P.M.
Mean of
observa
tions after
6 P.M.
Mean of
three pre
ceding
columns.
1862.
October ...
342
3*68
269
3*26
1863.
October ...
524
416
274
405
November
353
289
2*58
300
November
424
413
2*82
372
December
409
501
277
3*96
December
451
514
339
435
1863.
January ...
411
488
342
414
1864.
January ...
386
5*74
363
441
February
610
577
496
561
February*
4*78
4*97
316
430
March
628
510
502
547 1
March . . .
588
672
405
555
April
4*41
437
326
401
April
460
424
3'24
4*02
May ......
298
3*54
285
312
May
388
422
349
386
June
291
3*02
252
282
June
339
375
224
313
July
317
320
250
296
July
4*55
346
239
347
August ...
398
401
320
3*73
August . . .
404
372
239
338
September
398
441
318
386
A portion of the apparatus employed in the observations of atmospheric electricity
at Kew Observatory was erected in January 1861, but the observations which have been
* Observations on two days in February 1864 were out of range and have not been included; hence the
values here given are too low.
DR. EVERETT ON ATMOSPHERIC ELECTRICITY.
349
reduced commence with. June 1862, and extend to the end of May 1864, thus embracing
exactly two years.
The following description of the apparatus is, for the most part, copied verbatim from
a lecture delivered by the inventor, Sir William Thomson, at the Eoyal Institution,
May 18th, 1860, as reported in the Proceedings of the Institution*.
I„ The waterdropping Collector consists of an insulated copper vessel containing
water, which is allowed to flow out in a fine stream through a brass pipe projecting
through a hole in a windowframe on the west side of the Observatory into the open air,
which frame is, for still greater security, composed of ebonite. The nozzle of the pipe
is 11J feet above the ground, and 3 feet from the wall of the Observatory. The effect
of the flow of water is to reduce the copper vessel and its contents to the same electrical
potential as that point in the air at which the waterstream breaks into drops.
II. The dividedring electrometer, of which some of the internal parts are shown in
fig. 4, Plate XIX. consists of
(1) A ring of metal (A B) divided into two equal parts (CAD, C B D), of which one
is insulated, and the other connected with the metal case of the instrument and so with
earth.
(2) A very light needle (E) of sheet aluminium, hung by a fine glass fibre (H) and
counterpoised at G so as to make it project only to one side of the axis of suspension.
(3) A Leyden phial, consisting of an open glass jar, coated outside and inside in the
usual manner, with the exception that the tinfoil of the inner coating does not extend to
the bottom of the jar, which is occupied instead by a small quantity of sulphuric acid.
(4) A stiff straight wire (F E) rigidly attached to the aluminium needle, as nearly as
may be in the line of the suspending fibre, bearing a light platinum wire (K) linked to
its lower end and hanging down so as to dip into the sulphuric acid.
(5) A case protecting the needle from currents of air, and from irregular electric
actions, and maintaining an artificially dried atmosphere round the glass pillar or pillars
supporting the insulated halfring, and the uncoated portion of the glass of the phial.
(6) A light stiff metallic electrode projecting from the insulated halfring through the
middle of a small aperture in the metal case, to the outside.
(7) A wide metal tube of somewhat less diameter than the Leyden jar, attached to a
metal ring borne by its inside coating, and standing up vertically to a few inches above
the level of the mouth of the jar.
(8) A stiff wire projecting horizontally from this metal tube above the edge of the
Leyden jar, and out through a wide hole in the case of the instrument to a convenient
position for applying electricity to charge the jar with.
* This lecture, as reported in the Proceedings of the Royal Institution, was sent in the following year, with,
the photographic curves for four successive days, and an accompanying description, to the Philosophical Maga
zine, hut was not inserted ; and down to the present time no full description of the apparatus has been pub
lished, the most successful attempt that we have seen being the description of the Electrometers at Kew in the
*■ Engineer’ for August 9th of the present year.
350
DE. EVERETT ON ATMOSPHERIC ELECTRICITY.
(9) A very light glass mirror, about threequarters of an inch in diameter, attached
by its back to the wire (4), and therefore rigidly connected with the aluminium needle.
(10) A circular aperture in the case, shut by a convex lens, and a long horizontal slit,
shut by plate glass, with its centre immediately above or below that of the lens, one of
them above, and the other equally below the level of the centre of the mirror.
(11) A large aperture in the wide metal tube (7), on a level with the mirror (9), to
allow light from a lamp outside the case, entering through the lens, to fall upon the
mirror, and be reflected out through the plateglass window ; and three or four fine
metal wires stretched across this aperture to screen the mirror from irregular electric
influences, without sensibly diminishing the amount of light falling on and reflected
off it.
The divided ring (1) is cut out of thick strong sheet metal (generally brass). Its outer
diameter is about 4 inches, its inner diameter 2 \ ; and it is divided into two equal parts
by cutting it along a diameter with a saw. The two halves are fixed horizontally ; one
of them on a firm metal support, and the other on glass, so as to retain as nearly as
may be their original relative position, with just the saw cut, from rg to of an inch
broad, vacant between them. They are placed with their common centre as nearly as
may be in the axis of the case (5), which is cylindrical, and placed vertically. The
Leyden jar (3) and the tube (7), carried by its inside coating, have their common axis
fixed to coincide as nearly as may be with that of the case and divided ring. The glass
fibre hangs down from above in the direction of this axis, and supports the needle about
an inch above the level of the divided ring. The stiff wire (4) attached to the needle
hangs down as nearly as may be along the axis of the tube (7).
Before using the instrument, the Leyden phial (3) is charged by means of its pro
jecting electrode (8). When an electrical machine is not available, this is very easily
done by the aid of a stick of vulcanite, rubbed by a piece of chamois leather. The po
tential of the charge thus communicated to the phial, is to be kept as nearly constant
as is required for the accuracy of the investigation for which the instrument is used.
Two or three rubs of the stick of vulcanite once a day, or twice a day, are sufficient
when the phial is of good glass, well kept dry. The most convenient test for the charge
of the phial is a proper electrometer or electroscope, of any convenient kind, kept con
stantly in communication with the charging electrode (8). The gaugeelectrometer
described below was used for that purpose at Kew. Failing any such electrometer or
electroscope, a zinccopperwater battery of ten, twenty, or more small cells, may be very
conveniently used to test directly the sensibility of the reflecting electrometer, which is
to be brought to its proper degree by charging its Leyden phial as much as is required.
In the use of the dividedring electrometer, the two bodies of which the difference of
potentials is to be tested, are connected one of them with the metal case of the instru
ment, and the other with the insulated halfring. In the Kew observations of atmo
spheric electricity these two bodies were the earth and the waterdropping collector.
The needle being, let us suppose, negatively electrified, will move towards or from the
DE. EYEEETT OjY ATMOSPHEEIC ELECTEICITY.
351
insulated halfring, according as the potential of the conductor connected with this half
ring differs positively or negatively from that of the other conductor (earth) connected
with the case. The mirror turns accordingly in one direction or the other through a
small angle from its zero position, and produces a corresponding motion in the image of
the lamp on the screen on which it is thrown.
In the Kew apparatus, this image was thrown upon photographic paper, which was
drawn upwards with a uniform motion by clockwork, and a continuous trace of the
variations of electrical potential was thus produced. A zero line was at the same time
drawn by the image of the same flame reflected from a fixed mirror.
The curves of atmospheric electricity thus obtained are about 18 inches long; and
each sheet contains two. Each curve embraces a period of about twentyfour hours, the
paper having been regularly shifted or changed at about halfpast 10 a.m.
Generally speaking, the curves are distinctly traceable through the whole twentyfour
hours. The interruptions which do occur are owing, in some cases, to the spot of light
having moved too fast to leave a trace, in others to its having passed off the paper. As
regards this latter source of failure, it may be remarked that it is not detrimental to the
investigation of the law of diurnal variation. It merely does for us what General Sabine
found it necessary to do in combining magnetic observations, that is to say, it rejects
observations at times of unusual disturbance.
Specimens of the curves, of the actual size, are given in Plate XIX.
The ordinates (positive or negative) of the curves are to a close degree of approxi
mation proportional to the potential (positive or negative) of the air at the place of
observation, provided that the charge of the Leyden phial (3) be preserved constant;
and if this charge be allowed to vary, the ordinates will vary in simple proportion.
The charge was tested daily by the gaugeelectrometer, which we shall now describe,
and which is identical with the station electrometer used in my own Windsor observa
tions. Its external appearance is shown in fig. 1, Plate XVIII., and some internal parts
in figs. 2, 3, Plate XVIII. The same letters denote the same parts in all three figures.
It consists of
(1) A thin flintglass bell (fig. 1, Plate XVIII.) coated outside and inside like a Leyden
phial, with the exception of the bottom inside, which contains a little sulphuric acid (H).
The dotted line A A indicates the boundary of the tinfoil.
(2) A cylindrical metal case (shaded in fig. 1, Plate XVIII.), inclosing the glass jar,
cemented to it round its mouth outside, extending upwards about 1g inch above the
mouth, and downwards to a metal base supporting the whole instrument, and protecting
the glass against the danger of breakage.
(3) A cover of plate glass (C) with a metal rim, closing the top of the cylindrical case
of the instrument.
(4) A torsionhead (B, fig. 1, Plate XVIII) after the manner of Coulomb’s balance,
supported in the centre of the glass cover, and bearing a glass fibre (E, figs. 1, 2, Plate
XVIII.) which hangs down through an aperture in its centre.
352
PE. EYEEETT ON ATMOSPHERIC ELECTRICITY.
(5) A light aluminium needle (LL, figs. 1, 2, Plate XVIII.) attached across the lower
end of the fibre (which is somewhat above the centre of the glass hell), and a stiff plati
num wire (F, figs. 1, 2, Plate XVIII.) attached to it at right angles and hanging down to
near the bottom of the jar.
(6) A very light platinum wire (G, fig. 1, Plate XVIII.), long enough to hang within
oneeighth of an inch or so of the bottom of the jar and to dip into the sulphuric acid (H).
(7) A metal ring attached to the inner coating of the jar, bearing two plates (M, M,
figs. 1, 2, 3, Plate XVIII.) in proper positions for reflecting the two ends of the alumi
nium needle when similarly electrified, and proper stops (as O, fig. 3, Plate XVIII.) to
limit the angular motion of the needle to within about 45° from these plates.
(8) A cage (P P, figs. 1, 2, Plate XVIII.) of fine brass wire stretched on brass frame
work, supported from the main case above by two glass pillars (QQ) and partially
inclosing the two ends of the needle and the repelling plates, from all of which it is
separated by clear spaces of nowhere less than onefourth of an inch of air.
(9) A charging electrode (J, fig. 1, Plate XVIII.) attached to the ring (7) and pro
jecting over the mouth of the jar to the outside of the metal case (2), through a wide
aperture, which is commonly kept closed by a metal cap (K), leaving at least onequarter
of an inch of air round the projecting end of the electrode.
(10) An electrode (ST, figs. 1, 2, Plate XVIII.) attached to the cage (PP) and pro
jecting over the mouth of the jar to the outside of the metal case (2) through the centre
of an aperture. In order to dry any air which may enter through this aperture, a
hollow cylinder of pumice soaked in sulphuric acid is inserted in the leaden receptacle
(U, fig. 1, Plate XVIII.), through the centre of which the wire (T) passes, and the leaden
cover (V V) is pierced with a hole large enough for the wire to pass through without
contact. This cover has a depression (W) to receive the droppings of acidulated water
from the pumice.
This instrument is adapted to measure differences of potential between two conducting
systems, namely, as one, the aluminium needle (5), the repelling plates (7), and the inner
coating of the jar, and, as the other, the insulated cage (8). This latter is commonly
connected, by means of its projecting electrode (10), with the conductor to be tested.
The two conducting systems, if connected through their projecting electrodes by a me
tallic wire, may be electrified to any degree, without causing the slightest sensible mo
tion in the needle. If, on the other hand, the two electrodes of these two systems are
connected with two conductors, electrified to different potentials, the needle moves
away from the repelling plates ; and if, by turning the torsionhead, it is brought back
to one accurately marked position, the number of degrees of torsion required is propor
tional to the square of the difference of potentials thus tested.
In the ordinary use of the instrument, the inner coating of the Leyden jar is charged
negatively, by an external application of electricity through its projecting electrode (9).
The degree of the charge thus communicated is determined by putting the cage in con
nexion with the earth through its electrode (10), and bringing the needle by torsion to
DE. EVEEETT ON ATMOSPHERIC ELECTRICITY.
353
its marked position. The square root of the number of degrees of torsion required to
effect this measures the potential of the Leyden charge. This result is called the
reduced earthreading. When the atmosphere inside the jar is kept sufficiently dry, this
charge is retained from day to day with little loss, not more, often, than 1 per cent, in
twentyfour hours.
In using the instrument the charging electrode (9) of the jar is left untouched, with
the aperture through which it projects closed over it by the metal cap referred to above.
The electrode (10) of the cage, when an observation is to be made, is connected with
the conductor to be tested, and the needle is brought by torsion to its marked position..
The square root of the number of degrees of torsion now required measures the differ
ence of potentials between the conductor tested and the interior coating of the Leyden
jar. The excess, positive or negative, of this result above the reduced airreading,
measures the excess of the potential, positive or negative, of the conductor tested above
that of the earth ; or simply the potential of the conductor tested, if we regard that of
the earth as zero.
The mode of employing this instrument at Kew was to keep its Leyden phial (1)
always connected with the Leyden phial of the selfrecording electrometer by means of
a wire protected by an airtight tube, and to keep the cageelectrode (10) always con
nected with the earth.
Leadings of the gaugeelectrometer were taken daily at about 10 h 30 m a.m. and entered
in a book. Whenever the charge of the Leyden jars as thus tested was found to have
fallen too low, a fresh charge was given. The earthreadings of the gaugeelectrometer
were thus always kept between 245° as a lower, and 255° as an upper limit, or, to
speak more strictly, in the few instances in which these limits were exceeded, the obser
vations were rejected in the reductions. These readings, however, require to be cor
rected by subtracting the indexerror, which was carefully ascertained from time to time,
and never varied much from 230°. The corrected readings were therefore contained
between the limits 15° and 25°; and as the square roots of these numbers are as 1:13,
the weakest and strongest charges must have differed by about 30 per cent. This dif
ference, however, only affects the comparison of one day with another. The loss of
charge in twentyfour hours was from 1 to 3 per cent., and it is this loss only which
affects the diurnal curve. As the fresh charge was always given at about 10 h 30 m a.m.,
the disturbing effect is to be looked for in a sudden rise of the curve about this time, a
consideration to which we shall hereafter recur.
Since the erection of the Kew instruments the dividedring electrometer has under
gone considerable modification at the hands of its inventor. The flat ring (fig. 4) divided
into two segments is changed for a hollow box (figs. 5, 6) divided into four segments, of
which one opposite pair are connected with earth, and the other pair with the conductor
to be tested. This box encloses the needle, which is represented by. the dotted outline
in fig. 5, and projects symmetrically on both sides of the suspending fibre, thus obviating
the necessity for a counterpoise.
354
DE. EVEEETT ON ATMOSPHERIC ELECTRICITY.
The use of a gaugeelectrometer has been superseded by the introduction of a micro
meterscrew attached to one of the four segments, and regulating its distance from the
others. By diminishing or increasing this distance the sensibility of the instrument can
be increased or diminished at pleasure; and by attending to this adjustment as often as
may be necessary (say, once a day), the sensibility can be kept practically constant in
spite of variations in the charge of the Leyden jar*.
The retention of charge by the jar has been greatly improved by closing up the space
round the open electrode (as T, fig. 1) with vulcanite.
The portable electrometer employed in some of my Windsor observations has been
superseded by a smaller and at the same time more sensitive instrument, in which the
distance between two parallel plates, one of which is connected with a charged Leyden
jar and the other with the conductor to be tested, is varied at pleasure by a micrometer
screw so as to obtain a constant force of attraction between them. Equal differences of
potential, whether in the conductor tested or in the charge of the jar, correspond to
equal movements of the micrometerscrew, and the potential of the conductor can thus,
by comparison with an earthreading, be found by mere addition and subtraction.
The curves which have been measured and reduced comprise the two years commencing
with June 1862 and ending with May 1864. The method of procedure was as follows :r—
1. Ordinates were erected at every hour and half hour, careful attention being paid to
the times of commencing and ending, which were in every case indicated by entries made
on the photographic sheets by the Kew observers. In placing the ordinates, it was
assumed that each sheet had moved with uniform velocity through its whole length,
but it was not assumed that different sheets had moved with the same velocity ; in fact
the difference in the lengths of the curves, taken in connexion with the times of begin
ning and ending, showed that the velocities of different sheets must have differed by (in
extreme instances) about 5 per cent. As it was impossible therefore to use one time
scale for all the curves, about twenty different timescales were prepared differing by
small gradations one from another, and for the measurement of each curve that scale
was employed which suited it best.
2. The ordinates erected at the halfhours {e.g. halfpast one, halfpast two, &c.) were
joined by straight lines drawn by hand in such a manner as to give and take equal areas
as nearly as the draughtsman could judge.
3. The lengths intercepted by these joining lines on the hourly ordinates were mea
sured with a scale divided to millimetres, and were adopted as the mean heights of the
curve for each hour.
Whenever the curve for part of an hour was not traceable, a blank was left for that
hour, and whenever the curve was partly above and partly below the zeroline (showing
that the electricity was partly positive and partly negative), the algebraic mean was taken.
The measurements made in the manner above described were entered in a book in
* Still more recently a “ replenisher ” acting by induction and convection has been added, by means of which
the jar can easily be kept at a nearly constant charge.
DR. EVERETT ON ATMOSPHERIC ELECTRICITY.
;55
order of date, and from these entries the results shown in the subjoined Tables were
computed.
Table I. shows the mean electrical potential for each day, omitting those days on which
the number of blanks exceeded two.
Table II. shows the mean electrical potential for each of the twentyfour hours, month
by month, and the last line and column contain the means of the other lines and columns
respectively.
From the numbers in the body of this Table the curves (Plate XX.) have been drawn,
the first twelve lines of numbers being represented by the continuous curves, and the
next twelve lines by the dotted curves. These curves very clearly show a double maxi
mum and minimum, the principal maximum occurring about 8 p.m. in autumn and
winter, 9 p.m. in spring, and 10 p.m. in summer, and the secondary maximum about
8 a.m. in spring and summer and 9 a.m. in autumn. The principal minimum occurs
at 4 a.m. in spring and summer and 5 a.m. in autumn and winter. The curvature is
greater in the neighbourhood of the maxima than in the neighbourhood of the minima.
The mean diurnal curve at Kew (Plate XXI.) has been drawn by projecting the
numbers in the last line of the Table, the vertical scale adopted being twice as large as
for the twentyfour curves belonging to single months. Above this is placed the diurnal
barometric curve for Plalle, drawn from data contained in Kaemtz’s ‘ Meteorology,’ and
below is placed the diurnal electrical curve for Windsor, N.S., obtained by taking the
gross means of my own observations at each hour, observations from 2 to 6 a.m. being
entirely wanting*. The electrical curves for the two places are remarkably dissimilar,
both, however, having a maximum between 8 and 9 A.M.f The principal maximum at
Kew occurs between 8 and 9 p.m., and the principal minimum between 4 and 5 A.M.
The barometric curve for Halle bears a strong resemblance to the Kew electrical curve,
but is upwards of an hour later in phase. The slight rise in the Kew curve at 11 a.m.
is attributable to the fact that the Leyden jar was recharged at 10.30 a.m. By project
ing the numbers in the last column of the Table, we obtain annual curves of electricity
for two years. I have projected these so as to form one continuous curve, and along
* These means, together with the number of observations from which they are deduced, are as follows, in
units of “ station electrometer with second fibre.”
Hour
17 to 19
19 to 20
20 to 21
21 to 22
22 to 23
23 toO
0to 1
Number of observations . .
7
114
227
103
43
30
32
Mean
317
393
451
423
332
329
390
Hour
1 to 2
2 to 3
3 to 4
4 to 5
5 to 6
6 to 7
7 to 8
Number of observations . .
155
223
35
53
45
38
26
Mean
444
453
367
384
376
373
344
Hour
8 to 9
9 to 10
10 to 11
11 to 12
12 to 14
Number of observations . .
34
272
68
32
8
Mean
318
328
271
297
281
f At Windsor, in every month without exception, electricity wgs weaker at 10 p.m. than either at 9 a.m. or
2 p.m., hut the reverse of this rule would appear to hold at Kew.
MDCCCLXY1II. 3 D
356
DR, EVERETT ON ATMOSPHERIC ELECTRICITY.
with this I have projected the corresponding curve for Windsor, N.S. The Windsor
observations commence four months later than those of Kew and terminate three months
later, the time from October 1862 to May 1864 being common to both. In order to
ensure a fair comparison, as I have no means of comparing the units in which the
observations at the two places are stated, I have calculated the ratios of the several
monthly means to the annual mean, and have projected these ratios.
Inspection of the curves for the two places shows that they agree pretty well from
January to October, but take reverse directions from October to January, the Windsor
curve having a decided minimum in November, which is about the time of the principal
maximum at Kew. The annual range (as a fraction of the mean annual potential)
appears to be greater for Kew than for Windsor. The following are the ratios thus
plotted : —
Batio of mean monthly to mean annual potential.
At Kew.
At Windsor, N.S.
1862.
1863.
1862.
1863.
June
•770
June
•681
October
•832
October 1*033
July
•773
July ..
•643
November ...
•766
November ... *949
August
•836
August
•685
December ...
1010
December ... 1110
September ...
•845
September ...
•854
1863.
1864.
October
•981
October
1000
January
1057
January 1125
November ...
1600
November ...
1390
February ...
1432
February ... ?
December ...
1188
December ...
1460
March
1396
March 1416
1863.
1864.
April
1023
April 1026
January
1033
January
1226
May
•796
May 985
February ...
1333
February ...
1263
June
•720
June 799
March
1160
March
1375
July
•755
July *885
April
•920
April
•831
August
•952
August 862
May
1
•672
May
•549
September ...
•985
The final step in the reductions has been to express the variations of electrical poten
tial approximately by harmonic series. Both the diurnal and the annual variations have
been thus treated by calculating the values of the coefficients A 0 , A„ E„ A 2 , E 2 in the
formula
A 0 +Aj sin 360°+E^ +A 2 sin 360°+E 2 ),
T denoting twentyfour hours in the case of diurnal, and a year in the case of annual
variations ; and t denoting the time reckoned from noon in the former case and from the
middle of January in the latter.
The first step in this calculation consists in finding the values of P„ Q„ P 2 , Q 2 which
are connected with the abovementioned coefficients by the relations
Pi=Ai sin E 15 Q^Aj cos E 15 P 2 =A 2 sinE 2 , Q 2 =A 2 cos E 2 .
Commencing with the diurnal variations, we have the following values of the latter
coefficients for the twentyfour months of observation.
DR. EYEEETT ON ATMOSPHERIC ELECTRICITY.
35 /
First
year.
Second year.
P r
Qr
q 2 .
P r
Q,.
P 2 .
Or
June
•139
+
•185
+ •002
•241
•218
+
•114
•099
•328
July
•050
+
•053
+ •164
•456
•230
—
•066
+ •014
•364
August
•083
—
•Oil
+ •029
•507
•029
+
•203
•009
•298
September
•284
+
•409
•002
•490
•255
+
•378
•115
•467
October
•053
+
•835
•216
•315
+ •284
+
•586
•042
•310
November
— 264
+
•384
•036
•214
+ •109
+
•509
•166
•371
December
+ •028
+
•767
•058
•563
+ •297
+
•957
•017
•449
January
+ •006
+ 1060
•039
— *455
+ ■287
+
1271
•160
•297
February
•051
+
•743
•033
•582
•089
+
•630
•236
•561
March
—119
+
•008
•144
•516
•211
+
•441
•127
•565
April
•402
+
•056
•141
•535
•304
+
•311
•141
■440
May
•279
+
•006
•025
—•322
— 381
+
•044
+ •046
.95 
From these we derive the following values of amplitude (A } and A 2 ) and epoch
(Ej and E 2 ), regarding the former as essentially positive.
First
year.
Second year.
A
Ej.
a 2 .
e 2 .
1
Ay
E r
a 2 .
E g . j
June
•232
 36 53
•241
1
+ 179 24 j
•246
 62 28
•343
/ i
+ 196 45
July
•073
 43 26
•484
160 15
•240
105 53
•364
177 46
August
•084
 97 46
•508
176 45
•205
 8 5
•299
181 45
September
•498
 34 47
•490
180 17
•456
 34 2
•481
193 50
October
•837
 3 33
•382
214 21
•651
+ 25 53
•313
187 46
November
•466
 34 26
•218
189 39
•521
+ 12 3
•407
204 9
December
•768
+ 24
•566
185 55
1003
+ 17 15
•450
182 8
January
1060
+ 08
•457
184 39
1303
+ 12 43
•337
208 19 !
February
•743
 1 38
•583
183 14
•636
 8 1
•609
202 51
March
•119
 87 53
•536
195 37
•489
 25 36
•580
192 41
April
•406
 82 6
•553
194 45
•435
 44 19
•462
197 46
May
•279
 88 48
•323
184 30
•384
 83 29
•201
1 66 36
Year
•400
 20 36
•435
185 28
•452
 7 50
•395
192 50
The value of A 0 , or the mean electrical potential, is 2T4 for the first and 2T2 for the
second year.
It will be observed that the values of A x and E x are subject to much greater irre
gularities than those of A 2 and E 2 .
The values of P 15 Q n P 2 , Q 2 for the two years combined can be correctly found by
taking the means of their values for the two years, and the values of amplitude and
epoch (which can not be correctly found by taking the means of their values for the
two years) may be hence derived. The following Table has been thus computed.
3 d 2
358
DR. EYEEETT ON ATMOSPHERIC ELECTRICITY.
Two years combined.
Ai.
Ei
Hour of
maximum
from E r
a 2 .
e 2 .
Hours of maxima
from E 2 .
h
m
h
m
h
m
June
•232
50
4
9
20
•292
+ 190
3
8
40
20
40
July
•140
92
27
12
10
•395
166
40
9
27
21
27
August
•111
30
15
8
1
•398
177
59
9
4
21
4
September
•477
34
30
8
18
•483
187
1
8
46
20
46
October
•720
+ 9
16
5
23
•339
202
24
8
15
20
15
November
•454
 9
46
6
39
•310
199
6
8
22
20
22
December
•877
+ 10
39
5
17
•507
184
14
8
52
20
52
January
M74
+ 7
9
5
31
•389
194
45
8
30
20
30
February
•689
 5
25
6
22
•587
193
18
8
33
20
33
March
•279
36
15
8
25
•558
194
7
8
32
20
32
April
•398
62
28
10
10
•508
196
9
8
28
20
28
May
•331
85
40
11
43
•259
177
34
9
5
21
5
Year
•424
13
50
6
55
•413
189
0
8
42
20
42
The term involving and takes one maximum in the twentyfour hours, whose
times are given in the third column of the above Table. The term involving A 2 and E 2
takes two maxima which are always twelve hours apart. Their times are given in the
last two columns. In deducing the hours of maxima from the values of E x and E 2 , it is
to be borne in mind that the phases are earlier in proportion as the epochs are greater,
15° in the value of Ej and 30° in the value of E 2 corresponding respectively to differences
of an hour. It will be observed that the earliest and latest hours of maxima differ by
about seven hours in the case of E„ and by only one hour twelve minutes in the case of E 2 .
The values in the last line of the Table are derivable either from the last line of
Table II. or from the means of the values of P n Q,, P 2 , Q 2 . The amplitudes of the
diurnal and semidiurnal terms are, it will be seen, nearly equal. Practically, the hours
of electrical maxima month by month agree, within an hour or two, with those of the
semidiurnal term, and the hourly values for the average of the year, given in the last
line of Table II., show a still closer agreement. The diurnal term, without having much
effect on the times of maxima, causes one maximum to be much greater than the other.
It may be interesting to inquire into the connexion, if any, between electrical and
barometrical maxima, an inquiry which is suggested by the fact that the latter, like the
former, occur twice in the twentyfour hours. In default of the necessary barometric
data for Kew, I have compared the numbers in the last line of Table II. with the fol
lowing numbers which represent the mean heights of the barometer at Halle for all
hours on the average of the whole year, and are taken from Kaemtz’s ‘ Meteorology,’
page 248.
Barometric Heights at Halle (lat. 54° 29'), in millimetres,
750 plus the following numbers.
Noon.
329
l h .
311
2 h .
299
3 h .
289
4*.
284
5 h .
286
6 h .
291
302
8 h .
314
9 h .
324
^ CO
li h .
329
12 h .
323
13 h .
314
14 h .
3*05
15 h .
299
16 h .
299
17 h .
334
18 h .
312
19».
324
20 h .
337
21 h .
344
22 h .
346
23 h .
340
DR. EVERETT ON ATMOSPHERIC ELECTRICITY.
359
The value of E 2 derived from these numbers gives maxima at 10 h 28 m and 22 h 28 m , or
an hour and fortysix minutes later than E 2 for Kew electricity. About the same
amount of retardation can be roughly inferred either from inspection of the numbers
themselves, or from comparison of the curve representing them with the Kew electrical
curve (see Plate XXI.).
Thus far we have been speaking of diurnal variations. For annual variations, taking
the numbers in the last column of Table II. as our data, we have the following results : —
For the first year,
A 0 =2T4, A 1= 643, E 1= 107° 56', A 2 =080, E 2 =257° 31';
for the second year,
A 0 =212, Aj = 888, E 1= 102° 38', A 2 =030, E 2 =358°26';
and for the two years combined,
A 0 =213, Aj=*765, ^=104° 51', A 2 =040, E 2 =279° 19'.
The corresponding values for Windsor, N.S., derived from the numbers given in the
earlier part of this paper under the heading “ mean of three preceding columns,” are as
follows, the two years being combined, except for the months of February, August, and
September, which are taken from the first year’s observations alone, from defect of com
plete observations in the second year.
A 0 =392, Aj=725, E 1= 63° 43', A 2 =l014, E 2 =354° 29'.
In order to render these results for the two places more comparable, seeing that we
have no direct means of comparing the units employed, we shall, as we have already
done in the comparison of monthly means, take A 0 (or mean annual potential) as our
unit at both places. We thus obtain the following values for the two years combined : —
Kew . . . A 1= 359, ^=104° 51', A 2 =019, E 2 =279° 19',
Windsor . . A 1= 185, E 1= 63° 43', A 2 =259, E 2 =354° 29'.
This comparison brings out the astonishing fact that while at Kew the halfyearly
term is almost inappreciable, at Windsor its amplitude is actually greater than that of
the annual term. As regards phase, confining our attention to the annual term, Kew is
earlier than Windsor by about fortytwo days. The semiannual term at Kew is too
small and too fluctuating to admit of reliable comparison.
It must be borne in mind that the Kew reductions include wet as well as fine weather,
the only exclusions being those produced automatically by the spot of light passing out
of range or moving too rapidly to leave a trace ; whereas, in the Windsor reductions,
all observations taken during rain, snow, hail, sleet, fog, thunder, or lightning were
excluded. It is assumed, however, that this difference can go but a little way towards
explaining the great differences which we have detected in the electrical variations at
the two places.
I am happy to be able to state that preparations are now in progress for resuming the
photographic registration of atmospheric electricity at Kew with new apparatus contain
ing all Sir William Thomson’s most recent improvements.
360
DR. EYERETT ON ATMOSPHERIC ELECTRICITY.
1
; 110
138
138
211
■73
137
•89
•78
128
■96
1
11 slllSslSi SlsssS ;
1864.
1
i is s n ijssssssssssssi m  m
4
pH
s mi! j m \imm urn an mm
a
! H! !!!■!!! ! ISSS ! ! i*S*«*SI ! 15 !J
l
52i*!2I2!S2§tifSI2Sil!!SI2§i j i !
j?Ss2If i !i2222i!2!S2i2222I2 MM
s
S*8SsS2 i M m8S2SS22S Is Ml Hi! i
1
g m ig ; !gg$?gg?3? i? m i j j j
$
55s ■• m! 1 i i ; i !g=gggg  ils§§§
#
l?$ j ;s : ;s ;§ ;? M m?s
1
ha
• • • ^
1
I? 1 1 1? i§? g!2Sll§£22:r;§ ! ;!!£§ 1
1
3s :??? MSS :?• ;S3S2 ; i j S8 SS S3 S 2 5 23 3 S
^ * * * *
u
Sc
Mi;
1
mi! is iisii mnm miimm
I
mi llslllls is M M :522522m ! ; ;
1
i is m mi m mzimmmmmi
1
!1 !2”lSssJ!s!!Ss* I1SS2S2SW.M*
o
ft
siiisiii is i i isisi i i! : mm \ ;i \
i
m mu i i iiissss mm mm is 2 1
1862.
1
: igSSSS^g : i :33i??g MSS j : : : : :
■<
SS : : :SS MS? j???? jSSSS? :
iiggg iSsSzSS i2 i!2 i§ •;? i il!15SS!2
1
M M M MSS*? M M523SSS M M M j* !
1
^«eo^»o®^««o ;:S2322 j 5 c S2 © s .«e9 g  § « S; »«Oj 3
Table II.
DE. EVERETT ON ATMOSPHEEIC ELECTEICITY.
361
1
SSSSSSI
iltSSSSSisli
261
269
293
177
117
co
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CT CT O A 2h
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ggggggg
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£
s'
sSSSSsi
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88888
1
CT
1
h
=A!d— 1 + A'c cot 0 cos £ ;
10 =
khz — zVi
hz (iz U
h
= sin $ cos cos^ — j sin ^j
=L+M cos ^ f N cos 2£,
h^A'^— ^ sin 20,
where
378
LIEUT.GENERAL SABINE ON TERRESTRIAL MAGNETISM.
M=+^A'{c— a(c+a) cos 23},
N=£A'^sm23.
Also
= xhx \yty 4 z6z,
cos 2 3 cos £4^ sin £ J 4 sin 2 3 j
=P4Q cos £+R cos 2£,
where
P= + JA' cos 2, L, &c. at a basestation,
then at any other station at which the dip is 0, we have
L=
\ sin 20,
Qi
M=M,— Q, cot 23,4 sm 2 g cos 20
N =4k sin2< 
L i
P=P,4Li cot 3 , — —^r cos 23,
111 1 sm20, ’
Q=
R=
Qi
sin 20/
rj AL_( 1+cos2 S).
It will be observed that P, Q, R are abstract numbers, while L, M, N are angles the
numerical values of which depend on the assumed unit of angle. The values just given
may be used without modification if the angular unit be the angle subtended by the
arc=radius, or 57 0, 3. If the unit of angle be, say V, then in the expression for M we
must divide Q, by sin 1', and in the expression for P we must multiply L, by sin 1'.
As a check on the values of N and R, we may observe that we ought to have
^=+ri$bs =+ * A ' i(1 * )
=
in the notation of the Admiralty Manual ;
And that if we have the value of A'i(l — b ), or determined independently from
LIEUT. GENEEAL SABINE ON TEEEESTEIAL MAGNETISM.