1894-95.] Dr Muir on Prohlem of Sylvester’s in Elimination. 379 
ways, such is not necessarily the case ; and it is not difficult to 
indicate a general mode of procedure for obtaining them. 
From the original equations, by multiplying each in succession 
by Xy y, Zy we obtain nine equations in the ten unknowns 
x^y y^y z ^ ; xhjy y\ z^^x ; xy‘^y yz^y zx ? ; xyx . 
The tenth equation wanted for the purpose of elimination Sylvester 
obtained * by writing the original equations in the form 
(Ay - C'x)y + (Qa; - 0'y)x = 0 
(fiz- A!y)z + {Ry - A!z)y =0 - 
{Vz--B'x)z +{Cx-'B'z)x = 0^ 
and eliminating z, y, x, the residt being 
(Pa;-B'a:)(Ry-A'^)(Qa^-C'y) 
+ (Bz - A'y)(Ay - C'a;)(Ca; - B';^) = 0 , 
or 
2(CC'A' - QEB>2y + 1:(RB'C' - CAA>y2 + (aBC + PQB - 2 A'B'C>y;2 
It is easily shown, however, that this is not the simplest form of 
the tenth equation. For from the seventh equation, by multi- 
plying by A'B', we have 
AA'B'y% + QA'B'^a;2 - 2A'B'C'a!y^ 
and similarly BB'C% + BB'C'a:y2 - 2 A'B'C'ajye = 9 > 
and CC'A'a;2y + PC'A'ys2 - 2A'B'C'xy^ = 0 j 
and consequently 
tcC'AVy + tviB'C'xy‘‘ - ^M'B'O'xyz = 0 . 
From this and Sylvester’s equation by subtraction we have 
- 2QRB Vy - icAA V + (ABC + PQB + 4A'B'C>2/z = 0 , 
which, besides being much simpler, still retains the property of 
symmetry with respect to the interchange 
/a, B, C, y, z\ 
iQ> f t ] 
and is thus unique. 
Now, if we denote x^y y^, z^y a;2y, . . . . , xyz by 1, 2, 3, . . . , 10, 
it is seen that the ten equations respectively involve the variables 
* For another mode, see § 15. 
