355 
940] ON THE DEVELOPMENT OF (l+Tl 2 #)». 
and similarly 
r*(0, 1, 2, ..., r) = 1.2.3...rx £*(0, 1, 2, ..., r — 1); 
so that, omitting the common factor £*((), 1, 2, r — 1), we have 
m. m — n ... m — (r — 1) n.n* r - r ~ 1 divisible by 1.2.3 ...r. 
It thus appears that the fraction 
m . m — n ... m — (r — 1) n 
i. 2 7.. 7 ’ 
when reduced to its least terms, will contain in the denominator only products of 
powers of the prime factors of n; and it remains to show that multiplying this by 
n r it will become integral, or what is the same thing that 
1.2 .... r 
in its least terms will not contain in the denominator any prime factor of n. 
Considering in succession the prime numbers 2, 3, 5, ..., first the number 2, 
we see that in the product 1.2.3....r, the number of terms divisible by 2 is = , 
the number of terms divisible by 4 is = , that by 8 is = , and so on, where 
(q) denotes the integer part of ~, and so in other cases. Hence the product contains 
the factor 2, with the exponent + ..., which exponent is less than 
ry* ry* ry* 
2 + 4 + 8 + '" ‘ a< ^ 
is less than r, say it is less than (r). Similarly for the number 3, the product 
contains the factor 3, with the exponent 
+ 
+ ..., 
which exponent is less than 
ry* ry* ry* 
3 + 9 + 27 + ‘" ad in f‘ 
is less than say it is at most = (^r); and so it contains the factor 5 with an 
exponent which is less than \r, say it is at most = (¿r), and generally the prime 
factor p with an exponent which is less than —^ ^ r: say it is at most = ^ -^r'j. 
This is 
1.2.3 ... r = ^ 2< r) 3 (Jr) 5 (ir) ..., 
45—2