356 
[940 
m 
ON THE DEVELOPMENT OF (l +ri 2 x)n. 
where K is a whole number. Hence if n = 2 a 3^5* ..., we have 
1.2.3 ... r 
— X2 ro- < r >. 3 r P-(b r '>. 5 r v—(&' ] . 
and here for every prime number 2, 3, 5, ... which is a factor of n, that is, for 
which the corresponding exponent a, /3, y, ... is not =0, the exponents ra — (r), 
r[3 — {\r), rry — (|r), ... are all of them positive; and thus the fraction in its least 
terms does not contain in the denominator any prime factor of n; this is the 
theorem which was to be proved. 
Mr Segar’s theorem may without loss of generality be stated as follows: if 
/3, y, ... are any r — 1 unequal positive integers (which for convenience may be 
taken in order of increasing magnitude), then £* (0, /3, y, ...) is divisible by 
£i(0, 1, 2, ..., r- 1). A proof, in principle the same as his, is as follows: 
We have the determinant 
1, eft, a y , ... 
divisible by 
1, a, a 2 , ... 
b 
b 
c 
c 
viz. the quotient is a rational and integral function of a, b, c, ... with coefficients 
which are positive integers; hence putting a = b = c, ... =1, the quotient will be a 
positive integer number. Considering the numerator determinant, and for a, b, c, ... 
writing therein 1 + a, 1 +b, 1 + c, ... respectively, where a, b, c, ... are ultimately to 
be put each = 0, the value is 
= 1, l+^a + ^a 2 ..., 1 + y Y a + y 2 a 2 + ..., ... , 
b 
c 
where (3 1} /3 2> 
denote the binomial coefficients 
£ 
l ’ 
1.2 ’ 
&c.: 
attending only to the lowest powers of a, b, c, ... which enter into the formula, this is 
1, 
1, a, a 2 , ... 
1> @1, /3 2 
1, b, b\ 
1, 7i> 72 
1, c, c ! ,