264 Proceedings of Royal Society of Edinhurgh. [sess. 
A new Solution of Sylvester’s Problem of the Three 
Ternary Equations. By the Hon. Lord M‘Laren. 
(Read January 16, 1893.) 
The solution is effected by the method of Symmetric Functions. 
By a preliminary transformation * the equations of the original 
system are put into the form 
AX2-2C'XY + BY2 = 0) 
BY2-2A'YZ + CZ2 = o[ • 
CZ2-2B'ZX + AX2 = 0; 
The system is to he again transformed by dividing each equation by 
BY2, and putting f ® ^ ^ , 
Z JC 
= gwing 
+1 = 0 . . 
• • (1) 
. ^.e., , 
+1=0 . . 
• • (2) 
r-C3g + f2 = 0 . . 
• • ( 3 ) 
The roots of (1) and (2) may be denoted respectively by a and 
— ; h and • Now, as (3) has a root of | in common with (1), 
and has also a root of I in common with (2), if we substitute 
in (3) successively the four systems of values of ^ and and 
multiply together the four resulting expressions, one of the factors 
must satisfy (3), and therefore their product will vanish. 
The four factors are — 
e. 
/xb + 
a 1 \ 
and 
But since the 1st and 2nd factors (when cleared of fractions) are 
* The transformation is effected by dividing the original equations respec- 
tively by z^x^, and then putting X, Y, Z for . The 
X y z 
transformation used by Professor Tait in his paper (which I have seen in proof) 
would equally answer the purpose of this solution. 
