WHICH SATISFY GIVEN CONDITIONS. 
127 
x(hy-\-gz)=0, the two lines whereof intersect on the conic (1, 1 , 1 , X, p, vjx, y, z) 3 =0 ; 
and similarly, if g— 0, then h 2 -\-f 2 —2yjhf=0, or if h— 0, then f 2 +g 2 — 2\fg=0. As 
noticed above this system occurs twice. 
The second system is 
fgh= 0, (/ 2 +^+A 2 -2^-2^/-2^X1-^-^-v 2 +2xH 
+ ((p - + (M - v ) h f = °» 
or, as the second equation may also be written, 
+ 2^A(l — X 2 Xp — A) + 2^/(1 — y?)(v\— p) + 2fg( 1 — v 2 )(Kgj — 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 (\ — v 2 )-\-Ji\l—gj 2 )-{-2gh(gjv—'K)=-0, viz. we have here the line-pair 
x(hy+gz) = 0, in which the line %-fyz=0 touches the conic (1, 1, 1,X, 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. 
Annex No. 3 (referred to, No. 22). — On the conics which pass through two given points 
and touch a given conic. 
Consider the conics which pass through two given points and touch a given conic. 
We may take Z = 0 as the equation of the line through the two given points, and then 
taking the pole of this line in regard to the given conic and joining it with the two 
given points respectively, the equations of the joining lines may be taken to be X=0 
and Y=0 respectively. This being so, we have for the given points (X=0, Z=0) and 
(Y=0, Z=-0) respectively, and for the given conic 
aX 2 +6Y 2 +2AXY+cZ 2 =0 ; 
and since the required conic is to pass through the two given points its equation will be 
of the form 
wX 2 -f- 2xYZ-\- 2^ZX -f 2zXY = 0 , 
where (x, y, z, w) are variable parameters which must satisfy a single condition in order 
that the last-mentioned conic may touch the given conic. The condition is at once seen 
to be that obtained by making the equation 
(i a-\-\w)bc 
—(a+-kw){h+-hz) 2 
-fay 
— cX 2 z 2 
fi-2 7?xy(h-\-\z) = 0, 
