6 Mr. S. H. Burbury on the Law of Probability 



Part I. 



Proposition I. 

 12. Introducing b 2 , ... 6 n as auxiliary variables, 



+ 00 



<Kn • • • O =jj- ■ • ^ • ■<«. (J </>K ... O^i - ^v^ e v _1 . (3) 



— 00 



multiplied by a constant independent of r x ... r n . Here 



25 — r^ = 5! — r 1 ^ 1 + 5 2 --r2^2+ ■•■ + S n~~ r rPn'> 



and the limits of integration for each 6 are + go . Let the 

 right-hand member of the equation (3) be denoted by S. 



And in S substitute for e n ~° n nS < ~ l its equivalent 



cos s n -r n 6 n + ^/-l sin s n - rj n , 



and integrate according to 8 n from n =Ato O n = —A. That 

 reduces S to 



(J . . . de l de 2 . . . ae n _ Y (V . . <f>( Sl .. . o^i • • • *** 



the imaginary part evidently disappearing. This has to be 

 integrated now for s . Let 



(*»— O a =«» or *»=t +*■», and 



A, d# 



,/' 



Then make A infinite. The limits of x thus become + cc , 

 andS= ~ ' 



y ^A+---^i-r.. 1 <l. 1 ) VZT p sin * ^ ^ (5) 



— oo 



in which ^...jj is now by virtue of the equation 

 (s n — nJA = «£ a function of s x ... s n _ v x. 



