On Lunar and Solar Periodicities of Earthquakes. 457 



and if we had a sufficient quantity of material, it would be better to* 

 reduce the lengths of the intervals and to increase their number p,. 

 If this process is carried sufficiently far equations (2) become 



pa = n, 



\jpa x = cos /c^i + cos Kt 2 + ....-}- cos id n , 



ipbi — sm K^i + sin /ct 2 + ..... + sin ici„ (4) r 



where n is the total number of events and k stands for 27r/T, T being- 

 the whole length of the period, and t h t 2 , &c, the times of occurrence 

 of successive events. 



Equation (3) will become 



— {(cos k^ + cos td 2 + . . cos tct n y+ (sin /c^-f-sin tct 2 +. . -f sin «£„)*} * 



(5). 



The meaning of the expression on the right-hand side is best illus- 

 trated by means of a diagram. On a circle with centre at and 

 unit radius, take .points Pi, P 2 such that the angles between the 

 lines OPi, OP 2 and a fixed direction are kt u H 2 , &c. If OP 1? OP 2 

 represent forces of equal intensity but different directions, the right- 

 hand side of (5) gives the magnitude of the resultant force. As,, 

 according to hypothesis, the events may happen with equal prob- 

 ability at any time, every position on the circle is equally probable for 

 every point P. Under these circumstances it has been shown by 

 Lord Ray lei gh* in a paper " On the Resultant of a large number of 

 Vibrations of the Same Pitch and Arbitrary Phase," that the prob- 

 ability of the resultant having a value lying between s and s+ds is 



-e-*l n sds (6), 



n 



n being the total number of vectors combined. It is a simple matter 

 to pass from this result to the solution of our problem. 



From (5) and (6) it follows that the probability for the value of 

 nr^ao lying between \np and \n (p + dp) is 



^pe-k^dp (7) T 



and this is therefore also the probability that ri[a has a value inter- 

 mediate between p and p-\-dp. 

 The expectancy for t^/oo is 



;-f P w^= \/-^=~ (S) 



J Q n v n 



* 'Phil. Mag.,' vol. 10, p. 73 (1880, II). 



2e2 



