126 
PROFESSOR CAYLEY ON THE CURVES 
Conversely, if a given differential equation rational in x, y, ^ , and of the degree N in 
the last-mentioned 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 above-men- 
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 first-mentioned form is, it is clear, the one to be pre- 
ferred. 
Annex No. 2 (referred to, No. 17). — On the line-pairs 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 line-pair, and shall 
touch a given conic (1, 1, 1, X, vjx, y , zf= 0. The condition for a line-pair 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 <z, A, c, d being 
a— 2fgh, 
A = - Uf + / + A 2 - %Kgh - 2gJif- 2vfg), 
c — t((p'-*y+(^-p)A r +(¥-*') A )> 
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 contact-locus 
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<? 2 =0; the former of 
these is 
fgh— 0, ,/ 2 +/ + A 2 — 2kgh— 2phf— 2vfg— 0, 
which expresses that the intersection of the two lines of the line-pair intersect on the 
given conic; in fact the system is satisfied by/=0,/ 2 +A 2 — 2X^A=0, giving a line-pair 
