[649 
ADDITION TO MR GLAISHER’S NOTE ON SYLVESTER’S PAPER, 
“DEVELOPMENT OF AN IDEA OF EISENSTEIN.” 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xiv. (1877), 
pp. 83, 84.] 
The formula (11) [in the Note], under a slightly different form, is demonstrated by 
me in an addition [263] to Sir J. F. W. Herschel’s paper “ On the formulae investigated 
by Dr Brinkley, &c.,” Phil. Trans, t. CL., 1860, pp. 321—323. The demonstration is in 
effect as follows : let u denote a series of the form 1 + bx + cx 2 + da? + ..., and let 
(where i is positive or negative, integer or fractional) denote the development of the t’-th 
power of u, continued up to the term which involves x n , the terms involving higher 
powers of x being rejected; u°, u 1 , u 2 ,.., and generally u s will denote in like manner 
the developments of these powers up to the terms involving x n , or, what is the same 
thing, they will be the values of u* corresponding to i = 0, 1, 2,.., s. By the formula 
u i = 1 + \ (u — 1) + % 0 ^ ( u ~ 1) 2 + as far as the term involving (u—l) n , u { is a rational 
and integral function of i of the degree n, and can therefore be expressed in terms 
of the values u°, u 1 , u 2 ,.., u n which correspond to i = 0, 1, 2,.., n. Let s have any one 
of the last-mentioned values, then the expression 
i. i — \ .i — 2 ... i — n 1 
s.s-l ...2.1. —1. —2 ...—(№ — «)’ 
which as regards i is a rational and integral function of the degree n (the factor i — s 
which occurs in the numerator and denominator being of course omitted), vanishes for 
each of the values i= 0, 1, 2,.., n, except only for the value i = s, in which case it 
becomes equal to unity. The required formula is thus seen to be 
1. i — 2 ... i — n 
— (n — s)