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. 
