542 Proceedings of Royal Society of Edinburgh. [sess. 
efficients will be C x + C 2 + Now this is exactly what the 
expression 
Gf^aY 1 (^ab)^(2jabc)y^ . . . + C 2 (2a) a 2(2a6) fe (2«5c) 7 2 . 
when we put 
2a = 2a5 = 2a5c = . . . = 1 ; 
and this value each of these functions would have if instead of 
a, b, c 9 .. . the roots of the equation 
x n - x n ~ x + x n ~ 2 - . . . +(-) w l = 0 
were taken. 
This is the theorem of alternants which lies at the bottom of a 
curious theorem of Sylvester’s regarding “zeta-ic ” multiplication.* 
* Sylvester, “On Derivation of Coexistence,” Phil. Mag., xvi. pp. 37-43 
(1839). 
