42 
A smith’s prize paper, 1877. 
[645 
viz. the coefficient thus becomes 
1 
îi+a+J * 
Now a is supposed small in comparison with n, and the factors in the denominator 
have the logarithms 
a 2 
n ’ 
and 
a/ 
hence the denominator is = e n , and the final approximate value of the coefficient is 
Hence, converting as usual the sum into a definite integral, we have the sum of the 
2a +1 coefficients 
or, what is the same thing, 
a 
For the chance that the number of tosses lies between n + a and n — a, this has 
merely to be divided by 2 2n ; hence writing a = kn, the chance that the number may 
be between n (1 + k) and n (1 — k) is 
where observe that the integral, taken with the limits oo, — oo has the value \/(7r). 
Considering & as a given fraction however small, by increasing n we make 
k \/(ri) as large as we please, and therefore the integral, as nearly as we please 
= V(7r), or the chance as nearly as we please =1; and hence the complete and 
precise statement of the theorem, viz. by sufficiently increasing the number of tosses, 
the probability that the deviation from equality shall be any given percentage (as 
small as we please) of the whole number of tosses, can be made as nearly as we 
please equal to certainty. 
Further, restoring a in place of kn, the chance of a number between n + a and 
a 
U — ft IS