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



Each of these factors, e.g. sin (s —s p ')0 p , when integrated 

 between the same limits for s and for s ', gives the result 

 zero. Therefore every term vanishes in the integration for 

 s. . .s f except the product of the cosines, and 



jj . . . ds x . . . ds n Jf . . . &*l . . . dsj^ . . . s J<£0/ • • • O cos ( Sl - S& + 

 = jj . . . ds x . . .'ds n jj . . . ds/ . . . dsj4>{s l . . . sj <KV • • • O cos *i—*i0\ cos s 2 — s/^ 



•••cos^=^;^. 



As n becomes indefinitely great, this product becomes 

 indefinitely small, unless each of the factors cos a — s f is 

 equal to or nearly equal to unity. That condition is satisfied 

 for 0=0, but not if is any multiple of 7r, because if = 0, 

 cos s — s'0=l for all values of s—s'. But if is a multiple 

 of 7r, only for a particular value of s—s'. 



18. If this condition be not satisfied, a 2 + /2 2 , since it con- 

 tains the product of n cosines, which are not in general 

 nearly equal to unity, is, n being very great, indefinitely 

 small. Therefore a and /3, and therefore X, are indefinitely 

 small. We might fix limits between which 6 1 ... n respec- 

 tively must lie, say 



X between q x and —q x 



02 » 9 2 5 5 ~~Q2 



&c. 6 n „ q n „ -g n , 



in which q 1 ... q n are so small that all powers and products of 

 them above the second degree may be neglected. Then 

 unless ± ... 6 n lie within these limits, X is indefinitely small, 



and therefore also X<?~~~ ? ~ indefinitely small. If Q 1 ... 6 

 do lie within these limits, we may in evaluating X, and 

 therefore in evaluating 



neglect powers and products of X ... 6 n above the second 

 degree. 



19. In the expression 



Vn..^^...**..:**.-*" 1 ^ 



^L((.dd l ...deiid Sl ...ds n <f,(s l ...s x )e , . (12) 



