306 Proceedings of Royal Society of Edinburgh. [sess. 
Note on Dr Muir’s Paper, “A Problem of Sylvester’s in 
Elimination.” By Professor Cayley. 
(Received November 6, 1894.) 
I in part reproduce this very interesting paper for the sake of a re- 
mark which appears to me important, I write (a, h, c^f, g, h) in place 
of Muir’s (A, B, C, A', B', C'), and take as usual (A, B, C, F, G, H) 
and K to denote (he - /^, ca — ab — Jd, gli - a/, hf - bg, fg - ch) 
and the discriminant abc - af^ - bg"^ - eld + 2/^/l 
I then write 
U = - 2fyz + cy'^ , P = fx^ 4- ayz - hzx - gxy , L = hex^ + afyz - hgzx - dixy , 
Y = cx^ - 2gzx + , Q = gy‘^ - hyz + hzx - fxy , M = cay" - afyz + hgzx - dixy , 
AV = ay‘^ - 21ixy + hy ^^ , R = lid - gyz - fzx 4- cxy , N = ahd - afyz - hgzx 4- dixy . 
The equations U = 0, V = 0, W = 0, imply P = 0, Q = 0, R = 0, 
hut observe that P, Q, R are not the sums of mere numerical 
multiples of U, V, W ; we in fact have identically 
2y;sP = - -b y‘^Y -b W , 
^zxq= -y^^Y + z^W , 
Ixy'R — x‘^\J + yW - z'^W . 
If then U = 0, V = 0, W = 0, we have also P = 0, Q = 0, R = 0, 
and we can from the six equations dialytically eliminate x‘^, y‘^, z% 
yz, zx, xy^ thus obtaining a result, Determinant = 0, which is = 0 ; 
this is in fact Sylvester’s process for the elimination. 
But L, M, N are sums of mere numerical multiples of U, V, AV, 
viz., we have 
2L= -aU + Z^V-bcAY, 
2AI= aU~?>Y + cAY, 
2N= aU + iV-cW, 
so that the original equations U = 0, V = 0, AA^ = 0 are equivalent to 
and may be replaced by L = 0, AI = 0, N = 0. 
Muir shows that we have identically 
l.-fV =x(Ax + Hy + (Sz), 
M - p-Q = y(^^x -b By + F^) , 
N - 7iR = z((jx -b Fy -b Cz ) , 
