THEORY OF PROBABILITIES. 175 



Now let f $xxdx = K\ <j>x x*dx =. tf '; 



J o J o 



/.QO 



then, since / $xdx = 1, we shall have 



^ 



G = 1 - Jot 2 (m 2 - 

 H=a,(m S-ST) approximately. 

 Let w = 2j5T, then 



and thus = 1- ^a 2 2 (Aj 2 - /iT 2 ), 



J?=0; 

 and therefore, while a is very small, 



We have next to show that 2 (& 2 K z ] is a positive 

 quantity. 



Consider the definite integral 



-00 /- 



I I 



^0 ''O 



g xfdx dz ; 



it is necessarily positive, since every element is so, <f>x dx being 

 the expression of a probability, and therefore essentially positive. 



Expanding (s a?) 2 , we find for the value of this integral 

 k z -'2K z + 7c 2 or 2(tf-K z ). 



Hence & 2 - K*, and therefore 2 (k* - ^ 2 ), is positive. 



(This demonstration is due to Poisson, Con. des Terns, 1827.) 

 Returning to the value we have found for (a), we see that. 



we may in all cases represent (a) by cos aJe"^** 2 (**""**), since 



when a is very small, the two expressions tend to coincide; 



and when a is not so, both are sensibly zero, 2 (k* E?} being 



a large quantity of the order n. Consequently 



