372 
Proceedings of Boyal Society of Edinburgh, [sess. 
method, I might have given a solution founded on the same prin- 
ciple as these, but much simpler by reason of the fact that the 
particular form of the equations renders the actual solution of them 
unnecessary. For, as they stand, each equation may he viewed as 
an assertion regarding the sum of its roots; and, writing them 
explicitly in this form, we have 
1+f 
1+1 
i i 
= 2«3 
= 2^1 ■ 
= 2% 
and therefore by multiplication 
yj2 ^2 ^2 y-2 ^ 
= + ief‘ + - 4 , 
which is the desired result. 
My printed paper shows, however, that I had a more important 
object than this in view. 
4. The first fresh point to which I now wish to direct attention 
is the question of the occurrence of the eliminant in the form of a 
square. In elucidation of it we cannot do better than take a more 
general set of equations than Sylvester’s, and observe the way in 
which the eliminant of the former degenerates into that of the 
latter. 
Such a set of equations is 
Kif - 2C'xy + ~ ^ ) 
~ 2A'yz + ^ ^ i^) 
-2B'zx + Fz^ = 0 ) ^ 
where in the third terms P, Q, E, take the places of the A, B, C of 
the original set, and where each equation is derived from another 
of the set by changing the letters in accordance with the diagrams 
