458 



Prof. Schuster. 



The probability that the value of rjao exceeds p is 



- P e-Wdp = e-*V (9). 



2 



p 



If the higher coefficients r 2 = y/a 2 2 -\-b 2 i 1 &c, are treated in the 

 same manner, the same expression is found to hold. 

 Our final result may now be expressed as follows : — 



If a number n of disconnected events occur within an interval of time 

 T, all times being equally probable for each event, and if the frequency 

 of occurrence of these events is expressed in a series of the form 



I t — t x , t — t % , t-t n \ 



al l+/? 1 COS27r-^ r - + p 2 cos 4 tt + .... +p n cos 2pir — ^ — ) » 



the probability that any of the quantities p has a value lying between p 

 and p-\-dp is 



n 



and the expectancy for p is 



pe-^lHp, 

 2 



V ir\n. 



In proving this proposition it was assumed that the number of 

 intervals iuto which the period T is subdivided is very large, but this 

 condition is not essential. To suit accurately the process employed 

 by Mr. Knott, we should have to consider the vectors OPi, OP 2 , &c, 

 to be confined to fixed directions forming angles 2?r/p with each 

 other. But it follows directly from the method employed by Lord 

 Rayleigh that his results must apply to this case also, if p is a 

 multiple of four. It is further not necessary to enquire whether 

 (7) holds in the most general case, when p, for instance, is an odd 

 number, because the process employed by Mr. Knott and others is 

 justified only on the assumption that the number of intervals into 

 which the period is subdivided is so large that a further increase of 

 it would not alter the relative value of the coefficients of the Pourier 

 series. 



3. Unfortunately Mr. Knott has adopted a common but in my 

 opinion mischievous practice, which renders some farther reductions 

 necessary before we can apply expressions (7) and (8). Instead 

 of basing his calculations on the number of earthquakes which 

 took place in any particular lunar hour, he first takes over- 

 lapping means of the numbers put down for five consecutive hours. 

 This practice has its legitimate use, when it is desired to make 

 periodicities apparent to the eye by plotting down a series of 

 numbers which by themselves may be too irregular to bring out the 

 peculiarities whiuh it is intended to show. The averaging of a 



