1894 - 95 .] Prof. Cayley on Dr Muir's Pamper on Elimination. 307 
where observe that the first of these equations is 
(>2 - ayz){hz^ - 2,fyz + c^/) \ 
- (/?/2 - byz)(cx^ - 2gzx + az^) > = 2xyz(Ax + Hy + G^) 
- ) 
and similarly for the second and third equations. 
He thence infers that the elimination may be performed by elimi- 
nating X, y, z from the equations 
Ax -p Hy -p G;^ = 0 
Hx -p By + ¥z = 0 
G:r -P Fy -P C;s = 0 
viz., that the result is 
A, H, 
H, B, 
G, F, 
G 
F 
C 
= 0, that is K 2 = Q as before. 
The natural inference is that K being =0, the three linear 
equations in (x, y, z) are equivalent to two equations giving for the 
ratios x: y :z rational values which should satisfy the original 
equations U = 0, V = 0, W = 0: the fact is that there are no such 
values, but that K being = 0, the three equations are equivalent to 
a single equation : for observe that combining for instance the first 
and second equations, these will be equivalent to each other if only 
~ = ^ = that is, AB-H2 = 0, GH-AF = 0, HF-BG = 0, 
which are cK = 0, /K = 0, yK = 0, all satisfied by K = 0 ; and simi- 
larly for the first and third, and the second and third equations. 
It will be remembered that the true form of the result is' not 
K = 0 but K 2 == 0 ^ and this seems to be an indication that the three 
equations should be, as they have been found to be, equivalent to 
a single equation. 
The problem may be further illustrated as follows : instead of 
the original equations U = 0, V = 0, W = 0, consider the like 
equations with the inverse coefficients (A, B, C, F, G, H), viz.. 
B ;^2 _2Fy;^ +Cy2 = 0, 
C;^;2 ^2Gzx +Az‘^==0, 
Ay2 - 2H.x'y -p B^2 = q , 
