464 On Lunar and Solar Periodicities of Earthquakes. 



to the fact that tremors occur in groups (§5). The good agreement 

 between the phases of the periodicity disposes of that doubt. 



8. The expression for the expectancy deduced in § 2 may with 

 advantage be employed in similar investigations to decide the number 

 of events which it is necessary to take into account in order to 

 establish a periodicity of given amplitude. It follows from the laws 

 of probability that, to be reasonably certain, the amplitudes found 

 should be at least equal to three times the expectancy. Hence, if c 

 be the amplitude looked for, 



= 3^/^/11, or n = 9tt/c 2 = 28'3/c 2 . 



Thus, for instance, Mr. Knott deduces for his supposed lunar 

 period a range of 6 per cent., or an amplitude of O03. In order to 

 establish with certainty such an amplitude, it would be necessary 

 that the number of earthquakes taken into account should at least be 

 equal to 30,000, or over four times the number actually used in the 

 calculations. 



9. Tbe difficulty discussed in § 5 would seem to limit in many 

 cases tbe applicability of the results found. There are, indeed, in 

 nearly every case actually occurring in nature, certain regularities in 

 the manner in which events happen, and this regularity always favours 

 the higher values of the Fourier coefficients. As it will not be 

 possible to estimate in many cases the effect of the expectancy, some 

 other form of treatment will often be called for. The following 

 theorem will, I think, prove useful in these investigations : — 



Let y be a function of t, such that its values are regulated by some 

 law of probability, not necessarily the exponential one, but acting in 

 such a manner that if a large number of values of t be chosen at 

 random there will always be a definite fraction of that number 

 depending on t x only, which lie between ti and £i + T, where T is any 

 given time interval. 



Writing A = | y cos xtdt and B = 1 y sin ictdt, 

 and forming 



E = ^aF+W, 



the quantity R will, with increasing values of T, fluctuate about 

 some mean value, which increases proportionally to </T, provided 

 T is taken sufficiently large. 



If this theorem is taken in conjunction with the two following 

 well-known propositions, 



(1) If y = cos id, rl will, apart from periodical terms, increase 



proportionally to T ; 



(2) If y — cos \t, \ being different from /c, the quantity R will 



fluctuate about a constant value ; 



