«tew* 
244 ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. [406 
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 
(x = 0, y= 0, z— 0), we have to find (/, g, h) such that the conic (0, 0, 0,/, g, h\x, 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, p, v7[x, y, z') 2 — 0. The condition for a line-pair is that 
one of the quantities f g, h shall vanish, viz. it is fgh = 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 1 — (X, + 6f ) 2 — (p + 6g) 2 — (v + 01if -f 2 (X + 6f)(p + 0g)(v + 6h) = (a, b, c, d\6, l) 3 
suppose ; the values of a, b, c, d being 
a= 2fgh, 
& = — 3 (/ 2 + 9 2 + h 2 — 2Xgh - 2phf - 2vfg\ 
c = t ~ x )f + (vX-p)g + (Xp - v) h), 
d — 1 — X 2 — p- — v 2 + 2Xpv. 
Hence considering (f g, h) as the coordinates of the parametric point, we have the 
discriminant-locus a — 0, and the contact-locus 
a 2 d 2 + 4ac 3 + 4 b*d — 36 2 c 2 — Qabcd = 0, 
and at the intersection of the two loci, a = 0, b 2 (4bd — 3c 2 ) = 0, equations breaking up 
into the system (a = 0, 6=0) twice, and the system a = 0, 4bd — 3c 2 = 0 ; the former of 
these is 
fgh = o, / 2 +g 2 + h 2 - 2Agh - 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 f— 0, g 2 + h 2 — 2\gh = 0, giving a line-pair 
x(hy + gz) = 0, the two lines whereof intersect on the conic (1, 1, 1, A,, p, v][x, y, z) 2 = 0; 
and similarly, if g = 0, then h 2 +f 2 — 2phf — 0, or if h = 0, then / 2 + g 2 — 2Xfg = 0. As 
noticed above this system occurs twice. 
The second system is 
fgh = 0, (f 2 4- g 2 + h 2 — 2Xgh — 2phf — 2ifg) {\ — \ 2 —p 2 — ir + 2\pv) 
-1- [{pv — \)f+ (v\ — p)g -1- (p\ — v) hy = 0, 
or, as the second equation may also be written, 
f 2 (1 - p 2 ) (1 - v 2 ) + g 2 (1 - !/•>) (1 - \ 2 ) + h 2 (1 - \ 2 ) (1 - p 2 ) 
+ 2gh(l — X 2 ) (pv — X) + 26/(1 — p 2 ) (vX — p) + 2fg (1 — v 2 ) (Xp — v) — 0, 
which expresses that a line of the line-pair touches the conic; in fact the system is 
satisfied by f— 0, g 2 (1 — v 2 ) + h 2 (1 — p 2 ) -I- 2gh (pv —X) = 0, viz. we have here the line-pair 
x (hy + gz) = 0, in which the line hy + gz = 0 touches the conic (1, 1, 1, X, p, v\x, y, z) 2 — 0 
and the like if g = 0, or if h = 0. This system it has been seen occurs only once.