Fraser — Reduction of a Quartic Surface to a Canonical Form. 81 



The coefficient of 



X F X 2 4- . . . 



0. 



ZX^iJ = 2i 7.77T-X = 2i* -777 



/(A,) -' f'{K) 

 The coefficient of 



The coefficient of 



^ 4 K' Cr _ 



... gG^ +fF^ +(c- K')C^ = /(A,) = 0, 



.-. A^^C, = X,[eC, + (/G, +fF,'] = - cX' + . . . 



The coefficient of 



X.^F r^A .3 + . . .1 



-2y 



X^F 



^x = - 22,* ^^— = - 22,* 

 ' /(A.) 



>A/ + . . 



/(A.) J 



The coefficient of 



.., = - .V^-^-'AA,) ^ - .V)^§ = - 2...i^t^ = - U. 



Hence 



^ 4 A^Cv 



— .^1 -77; 



/'(^.) 



{F^+G^/)z 

 xy + X^z 



and 



7'(A.) 



«y 



= a;2/ + 32[(«, J, c,f, g,h) {x, y, s)^], 

 {F^ + 6^^y): 



a 



+ A^z^ 



^ 0. 



The bicii'culai' quartic might be treated in precisely the same 

 manner, s = denoting tlie line at infinity in the plane, and x -¥ iy 

 written for x, x - iy written for y. But it can also be treated 

 directly, thus — 



{x? + y-y + ax^ +hy- -\- c + 2gx + 2fy = 



denotes its equation. 



