178 THEORY OF PROBABILITIES. 



and, lastly, that of t" is 



1 2 2 1 



" 4 (n - 1) + 4 (n - 1) 1 . 2 . 3 (n - I) 2 " 4 (n - 1) 



n i 



or 



1.2.3.4 (n-l) 2 4(n-l) ! 

 1 



1.2 12(-1)J 4 (n-l) 3 ' 



Now if, in forming the approximate expression, we reject all 



{t } p 1 

 -7-r^rr q , where q is different from zero, 

 vWJ w 



*. e. if we look on .. . as a quantity all whose powers are to 



be retained, except when divided by any power of n, the 



/ t \ n ~ l 



value of e~M 1 H --- -J may be taken as equal to 



t _L ** _* 



2(w-l) + 1.2*4(n-l) 2 



which, as similar results would have been obtained had we 

 pursued the investigation farther, might be shown to be equal to 



ff 



c -2(n-l) ^ 

 1 -J 2 



and thus P = -77- N e~ 2 ("- 1 ) dt, 



P is the probability that T is equal to n I -f t : writing 



^ 

 t + 1 for t in and reducing, we find that, within the 



limits of the approximation, it may also be assumed as the 



f f 

 probability that t is equal to n+t: also [ = - <L-P"> 



and thus 



which, as in our case, k* = 2 and K= 1 is precisely equivalent to 

 the result deduced from the general. formula (p). 



The legitimacy of some parts of the preceding approximation 

 may be questioned; as quantities which are neglected may, 



