406] 
ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
255 
The coefficients of the powers 16, 15, 14, 13 of 6 all vanish, so that this is in 
fact an equation of the twelfth order (*\6, 1) 12 =0; and putting, as usual, 
(bc-f 2 , ca-g 2 , ab-h 2 , gh-af hf-bg,fg-ch) = (A, B, G, F, G, #), 
the equation is found to be 
- 4cA 6 12 
+ 30c# d u 
- 36c£I 
6> 
+ 16fA 
- IOcG ] 
+ 40g A f 
+ 20b A \ 
- 60fB I <9 8 
- 90gH) 
+ 
72 hA\ 
+ 
W 
- 
225#) 
+ 
40a^l' 
- 
1305# 
+ 
+ 
40fF) 
+ 
33hB \ 
+ 
2 bG 
108«#] 
+ 45 aB \ 
- 20fC 6* 
+ lOhGJ 
+ 5 liF 
e s 
& 
+ 20aG 
- 4bC' 
- !2aFj 
- 5hC e 
- aC =0, 
where the form of the coefficients may be modified by means of the identical 
equations 
(A, #, 0$a, h, g) = K, 
{H,B,F\ „ ) = 0, 
(G,F,C& „ ) = 0, 
(A, #, G^h, b,f) = 0, 
CH,B,F$ „ ) = K, 
(G,F,C\ „ ) = 0, 
(A, H, G\g,f, c ) = 0, 
(#, B, FI „ ) = 0, 
(G,F,C$ „ ) = K. 
There is consequently a conic answering to each value of 6 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, K$x, y, zf = 2 (Xx + /iy + vz) (X'x + y!y + vz), 
then, writing for shortness 
gv' — fiv, vX 1 — vX, X/i — X'fi = X, Y, Z, 
we have 
(A, B, G, F, G, H) = (X 2 , Y 2 , Z 2 , YZ, ZX, XY).