of a Number of Unit Vibrations. 503 



where 2sb = a, and ultimately s will be made infinite. The 

 question is — What is the chance t'uat the sum of the distances 

 of n points shall be equal to lb, where / is a positive or 

 negative integer ? On examination it appears that the 

 combination follows the same laws "as the addition of 

 indices in the successive multiplication of the polynomial 



e -rs9 + e -i(s-V.9 + e -«s-2)e + . . . . + ^-^ + ^D» + ^ 



by itself, supposing the operation repeated w — 1 times. 

 And therefore the number of combinations required will be 

 the coefficient of e (which is also the same as the coefficient 

 of e~ lle ) in the expansion of 



The number of combinations required is therefore the same 

 as the term independent of 6 in the expansion of 



or the same as the term independent of 0/' when 



cos 16 {1 + 2 cos 6 + 2 cos 20 + .... + 2 cos sd\ n 



is expanded and arranged according to cosines of multiples 

 of 6. By summing the series and application of Fourier's 

 theorem this term is found to be 





ia f sin ±(2s+l)0\» ja , Q . 



cos 16 -s . , a J > cW. ... (9) 



L sin \v J v ' 



This is the number of combinations which gives rise to 

 a sum equal to I, and in order to obtain the probability 

 of I it must be divided by the whole number of combinations 

 equally probable, that is (2s -J- 1)*. What we have to 

 consider is accordingly the value of 



7r(2* + l) n Jo I sin±0 / 



In their discussion, Laplace and Airy regard both n and * 

 as infinite. Here it is proposed to make s infinite, so as to 

 attain a continuous distribution of the points, but without 

 limitation upon the value of n, which may be any integer. 

 If, as before, a denote the sum of the distances, 



o-=Z6=Za/2.s. 



When s is very great, sin sO alternates with great rapidity,. 



