300 Proceedings of Royal Society of RdinhurgJi. 
A Problem of Sylvester’s in Elimination. 
By Thomas Muir, LL.D. 
(Eead April 4, 1892.) 
1. The problem in question with its solution appeared in the 
Cambridge Mathematical Journal, vol. ii. pp. 232-236, being given 
by Sylvester to show how his recently discovered “dialytic” method 
might be applied to ternary quadratics. The given equations are 
Ay^-2C'xij + -Bx^ = 0\ 
Bz2 - 2A!yz 4 - = 0 > (a) 
Cx^^-^^'zx +Az^ = 0) ; 
and Sylvester’s solution consists in deducing from them three other 
equations 
Cz^ + Cxy - A!zx - B'y^ = 0 
A'x‘^ + Ayz - 'Exy - Czx = 0 V (^8) 
B'y2 + ^zx - Cyz - A!xy = 0 j ^ 
and then eliminating x^, y^, z^, yz, zx, xy ; the result being 
. c 
B 
-2A' 
• 
c . 
A 
-2B' 
B A 
-2C' 
A' . 
A 
-C' 
-B' 
. B' 
. 
-C' 
B 
-A' 
C' 
-B' 
-A' 
C 
This determinant on being expanded, not without considerable 
trouble, is found to be equal to 
2(ABC + 2A'B'C' - AA'2 - BB'2 - CC'2)2 
2 
A C' B' 
C' B A' 
B' A' C 
2 
