ON THE DEVELOPMENT OF (1 +n*x) n . 
[From the Messenger of Mathematics, vol. xxii. (1893), pp. 186—190.] 
It is a known theorem that, if — be any fraction in its least terms, the 
n J 
m 
coefficients of the development of (1 +n 2 x) n are all of them integers, or, what is the 
same thing, that 
to. to — n ... m — (r — 1) n „ 
1 . 2 ... r 
is an integer. The greater part, but not the whole, of this result comes out very 
simply from Mr Segar’s very elegant theorem, Messenger, vol. xxii. (1893), p. 59, “ the 
product of the differences of any r unequal numbers is divisible by (r —1)!!” or, as 
it may be stated, if a, /3, 7, ... are any r unequal numbers, then £*(a, /3, 7, •••) is 
divisible by £^(0, 1, 2, ..., r — 1). 
In fact, writing r +1 for r and considering the numbers 
m + n, n, 2n, 3n, ... (r— l)n; 
then neglecting signs 
£* (a, /3, 7, ...) is = m . m — n ... m — (r — 1) n, 
x In . 2n ... (r — 1) n, 
x In . 2n ... (r — 2) n, 
xl)i, 2n, 
X In, 
which is 
= m . m — n ... m — (r — 1) n x n^- r 1 x £» (0, 1, 2, ..., r — 1),