on. [647 
648] 
57 
648. 
ALGEBRAICAL THEOREM. 
ch node, or 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xiv. (1877), p. 53.] 
surface, its 
i the inter 
file tangent 
e has thus 
I wish to put on record the following theorem, given by me as a Senate-House 
Problem, January, 1851. 
If {a + /3 + y + ...}* denote the expansion of (a + /3 + y + ...y, retaining those terms 
Na a /3 b y c ... only in which 
b + c+ d + ...$>p— 1, c + d +...$> p — 2, &c., &c., 
then 
x n = {x + ct) n — n {a} 1 (x + a + /3) n_1 + \n (n — 1) [a + /3} 2 (x + a + /3 + y) n ~ 2 
-^n(n-l)(n-2){a + l3 + y} 3 (x+a + l3 + y+8) n - 3 + &c. 
The theorem, in a somewhat different and imperfectly stated form, is given, Burg, 
Crelle, t. I. (1826), p. 368, as a generalisation of Abel’s theorem, 
(x + a) n = x 11 + na (x + /3) n_1 + \n (n - 1) a (a - 2/3) (x + 2¡3) n ~ 2 
+ ^(n — 1) (n — 2) (n — 3) a (a — S/3) 2 (x + 3/3) 2 + &c. 
c. x. 
8