150 
Proceedings of Boyal Society of Edinburgh. [sess. 
A Problem of Sylvester’s in Elimination. 
By E. J. Nanson, M.A. Communicated by Professor Chrystal. 
(Read December 6, 1897.) 
1. Denoting by A, B, C, F, G, H the co-factors of a, b, c,f , g, h, 
in the determinant 
a li g x 
h b f y 
g f c z 
x y z . 
the problem in question is to eliminate x, y , z from the three 
equations 
A = 0, B = 0, C = 0 (1) 
and it has been discussed by Sylvester,* * * § Cayley, f Muir, { Tait,§ 
and Lord M‘Laren.|| 
Sylvester deduces from (1) this new system 
F = 0, G = 0, H = 0 ... (2) 
and hence finds the dialytic eliminant 
. c b -2/ . 
c . a -2 g . 
b a . . - 2 h 
f . .a -h - g 
. g . -h b - f 
• • K-g -/ C 
(3) 
the expansion of which (an obvious error being corrected) he 
states to be 
abc& = 0 
* Cam. Math. Journ ., ii. p. 232. See also Muir, Theory of Determinants, 
part i. p. 231. 
+ C. M. P., 198 ; Proc. E.S.E . , xx. p. 306, 
+ Proc. R.S.E. , xx. pp. 300, 371. 
§ Proc. R.S.E., xix. p. 131. 
|| Proc. R.S.E., xix. p. 264. 
