Mr Babbage m a Theorem in Prime Numbers, 4*T 
n being a whole number, now it is well known, that the suns 
of the squares of the coefficients of a binomial whose index is w, 
is equal to the coefficient of the middle term of another binomial 
whose index is ^^e have therefore the equation 
Sn.Sw — 1. — 2.... w-l-1 
+ 
+ (>) + 
or 
{ 
W-f- 2...271 1 
3,> + 
But the quantities their equals#- 
n n.n 
T’ 
1 
1.2 ’ 
nn — 1. 
&c. 
n.n- 
1...2 
1.2.3. .w—1 
are all divisible by n when n is a prime, but they are not all di- 
visible by n when n is not a prime ; and, since the quantities on 
the right side consist of the sums of the squares of these, it i# 
divisible by w®, and consequently 
2 
7i-\-l.n-{-2....2n — 1 
1.2...7Z— 1 "S 
is always divisible by n in the same circumstances, as 2 cannot 
be divisible by n except ?i=2 we may omit that factor. The 
same theorem may also be put into the following form : 
n — 1 
2 
1.3.5....27^' 
1.2.3...W 
-i.2— 1 I 
is always divisible by ti ^ when w is a prime : This is immediate- 
ly deduced from the former by the equation 
714-1. 7^4-2....27^=::2n.l.3.5....27^ 1. 
Several theorems of a similar kind may be deduced from the in- 
vestigations of Euler, relative to the properties of the co-effi- 
cients of a binomial. See the Acta Acad. Sclent. Petrop. 1781. 
Retaining the notation already employed, it is there shewn, 
that when p n and q are any whole number, we have 
