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



increment (^ 2tt) of x, the increment of (j){s 1 ... s n ) is infini- 

 tesimal. Therefore in the integral 



i 



*l ■ ■ ■ sin a 



^""^-i n ^ing ^ss than 2-77, <j>(s 1 :.. s n ) may be regarded as 

 constant, and therefore 



J (/)( 5l ... 5jl ) Sm ^^ = 

 "V-i * 



for each integral value of ^ greater than unity. Therefore 



J" 30 T, . sin^ 7 f* ., N sin A' 



But in this integral, since z is less than 7r, the range of # is 

 less than 2tt, and $ (^ ... s n ) may be treated as constant, 

 having the value which it has when #=0, that is when 



(*»-O Aa =0, or ^=r„, and <£0i ... = +(*i - WJ' 



We have then 



jj...^ 1 ...^jj...^ 1 ...o^ 1 --^^ 1 



=zJJ... <^ ... rf^jj"... <K* ... w^i ... ^i^ 1 '"' 8 VZI - W 



15. As the result of the two integrations for n and s n , 6 n 

 has disappeared, and s n has been replaced by r n . And by suc- 

 cessive double integrations in this way 



rr rr ss-r e v - 1 



(j...^!...^ ...CJ>(s 1 ...S n )ds 1 ...dS n e 



is reduced to Z n <^>(r 1 . . . rj ; or 



♦fa -0= ~ (T rf«i •■• M* jJ*-*(<» - 0*i ••• 'V 



ss-^ev-i 



i (7) 



1 . 



and -^ is a constant independent of r Y ... r ? . 

 Proposition I. is thus proved. 



