174 THEORY OF PROBABILITIES. 



x^ ... x n _ l , which do not make T x l ... x n _ l negative. Thus 

 we have 



P= dx n f . . . f^x, fax 9 ...<f> n (T-x l -... x^) dx^.. dx^ . 

 Now, by Fourier's theorem, 



i f r 



4> n (T-x l -...x^_ 1 ) = --l da <t> n x n cosa(T-x l -...x n _ l -x n )dx n , 



" J o J 



T Cj . . . #_! being supposed to lie between and GO ; for all 

 negative values of this quantity, the second member of the 

 equation is equal to zero. Consequently, all the integrations 

 may now be taken from zero to infinity; and thus, since as 

 T and x n vary together, dT=dx n 



^rp /-oo / ,.00 



P -- I doi I dx l ... I dx^x^ . . . <f> n x n cos a ( T- Jsc). 



7T J Q J Q J Q 



Now it may be shown that the greatest value of 



r r 00 



dx l ... dx^x^... (f> n x n cosa (T-2x) ... (a) 



^o J o 



corresponds to a = 0, and is, therefore, unity ; and that when 

 n is large (a) diminishes rapidly as a increases. Consequently 



/.oo 



the value of I (a) da. depends, when n is very large, on the 



'o 



elements for which a is very small. This consideration enables 

 us to employ an approximate value of (a). 



Let T=t + m, m being a disposable quantity ; then 



/< /" 



(a) = cos at I dx l ... I dx^x^ . . . (f> n x n cos a (m 

 Jo J o 



-CO ,0, 



-f sin cat I dx l ... I dx n fax^ . . . <f) n x n sin a (m S#), 

 ^o Jo 



which may be thus written, 



(a) = cos OitGr + sin CLtH. 



In order to obtain approximate values of 6r and H, expand 

 cos a (m 2x) and sin a(m- 2#) ; they become respectively, 

 a being very small, 



1 - Jot 2 (m - 2x)* and a (m - 



