456 



Prof. Schuster. 



ti t 2 £3 . . . . tp 



ti t% t$ • • • • tp 



T x T 2 T 3 .... T p 



Ti, T 2 , &c. represent the sums of the vertical columns. In the case 

 discussed in the first part of Mr. Knott's paper the intervals are 

 hours, and p is taken to be 25, i.e., approximately equal to the 

 interval between two successive meridian passages of the moon. 

 Ti would therefore represent the number of earthquakes which have 

 happened within an hour after the meridian passage of the moon. 

 The numbers T may be expressed by a periodic series of the form 



S = ao + ax cos 0-\-a 2 cos 20 + . ... + a p cosp0 



+ 5 1 sin0 + &2sin20+ +b p _ 1 sm(p + l)0 (1), 



where S = Ti if we substitute = 27r/p, and generally S becomes T ? 

 by the substitution = 27rqjp. 



The coefficients are determined by a well-known process, which 

 tuves 



pa = T L -fT 2 + + T 7 



,1 



\pa x = TXCOS0 + T2COS20+ -fT^cos^fl > e = 27r lp ( 2 )- 



\pb x - T 1 sin0 + T 2 sin20+ sinyflj 



The amplitude of the first periodic term would be r x m y/ a^ + b-^. 



It is seen that o is equal to the mean value of all the quantities T 

 or to s times the mean values of all the quantities t. 



From the above equations it follows that — 



f? = (T x cos 9 + T 2 cos 2 6 + .. + Tp cos p9)' 2 + (T x sin 9 + T„ sin 2 9 + . . -r T p sin pd)- 

 4^7 (T 1 + T 2 + ..T^ 



(3). 



We may take the quantity p == rjcto as a measure of the periodicity 

 corresponding to p intervals. Our problem now is this : " What is the 

 probability that p should lie between any two assigned values p l and /> 2 , 

 on the supposition that the events are all distributed at random.''' The 

 problem may be put into a more general form. The events, like the 

 earthquakes in Mr. Knott's paper, have all been put into the same 

 compartment if they happened within certain interval of time (an 

 hour in this case), no matter whether they happened at the beginning 

 or at the end of that interval. This simplification is introduced only 

 for purposes of more easy arithmetical calculation. Theoretically, 



