NOTE ON A GENERALIZATION OF A BINOMIAL THEOREM. 
[From the Philosophical Magazine, vol. vi. (1853), p. 185.] 
The formula (Grelle, t. I. [1826] p. 367) for the development of the binomial (x + a) n , 
but which is there presented in a form which does not put in evidence the law of 
the coefficients, is substantially equivalent to the theorem given by me as one of the 
Senate House Problems in the year 1851, and which is as follows:— 
“If {»+/3 + 7...}^ denote the expansion of (a + (3 + y ...) p , retaining those terms 
Na a /3 b y e 8 d ... only in which h + c+ d ... is not greater than p — 1, c+d+.. is not greater 
than p — 2, &c., then 
x 11 =1 (x + a) n 
- - {a} 1 (x + a + /3) n_1 
+ {a -4- /S} 2 (x + a + /3 + y) n ~ 2 
— n ( n i ^[a + /3 + 7} 3 (x + a + fi + 7 + 8) n ~ x . 
+ &c” 
The theorem is, I think, one of some interest.