Mr Campbell, The study of discontinuous phenomena. 119 



In calculating these means we have assumed that a ' very 

 large number' of sets of s trials are taken. It is desirable to 

 inquire how large the number must be in order to reduce the 

 probable error within any desired limits. The problem is very 

 similar to that of determining the probable error of the calculated 

 probable error of a set of observations. If s is very large, it is 

 well known that the expression given for the probability of a 

 'deviation' x may be reduced by Stirling's theorem to an exponential 

 form, so that the probability that the value of x lies between x-^ 

 and x^ + dx is 



Q-K^xi^ ^g/,^ where h = 



s/ir ' '^2pqs 



Accordingly the probability of a given set of o- values of x, Xi, x^, 

 X3... such that Xi' -\- x^ + x^ + ... xj^ = ara^ is 



h X"' 



~i- dx] e-'^'-<^»'. 



Vtt / 



We shall find that our observations give us the value of 

 aa^ = liX^. h is then chosen so as to make the probability of the 

 occurrence of this value of X^c^ a maximum : we find 



h"^ = TTFT— or a:;2 = — - = spq, as before. 

 zzx^ 2h 



Accepting these values, we can find the probability that h has 

 the value h + uh: this probability is 



\ Vtt / 



f h \'"' 

 where G = (-^ dxj e-^'-''"'\ 



Hence the ' measure of precision ' of A is \/n. In order that the 

 probable error may be less than 1 °/^, \/n must be greater than 

 100 or n greater than 10^ In order to attain this degree of 

 accuracy we must take 10* sets of the series of s trials. We have 

 assumed that s is large, but since the expression for the probability 

 will be of nearly the same form whatever the value of s and since 

 we require only a rough value for the measure of precision, we 

 may apply the result to all values of s such as are likely to occur. 



I 3. Let us now apply this theorem to such cases as are likely 

 to occur in physical investigation. Suppose that a very large 

 number of events NT happen in a time T. If we fix our attention 

 on a period t within T and small compared to it, the chance that 



any one of these events will happen within t is „: it must be noted 



VOL, XV, PT, II, 9 



