TRANSACTIONS OF SECTION A. 515 



Conncctious between the moon and earthquakes, or between snnspots and rain- 

 fall if proved to exist, would form examples of such relationship. The question to 

 be decided in these cases would be : Is there a lunar period of earthquakes, or an 

 eleven years' sunspot period of rainfall ? 



Everyone familiar with Fourier's analysis knows that there is a lunar or sun- 

 spot, or any other period in any set of events from volcanic eruptions down to the 

 birth-rate of mice; what we want to find out is, whether the periodicity indicates 

 a real connection or not. Let us put the problem into its simplest form. Take 

 n balls, and by sonie mechanism allow them to drop so that each falls into one of 

 m compartments. If finally they are equally distributed each compartment would 

 hold nlm balls. If different compartments contain a difi'erent number of balls, we 

 may wish to find out whether the observed inequality is sufficient to indicate any 

 preference for one compartment, or how far it is compatible with equality of chance 

 for each. If we were able to repeat the experiment as often as we liked, we should 

 have no difficulty in decidinir between the two cases, because in the lonp: run the 

 average number received by eacli compartment v/ould indicate more and more 

 closely the extent of bias wliich the dropping mechanism might possess. But we 

 are supposed to be confined to a single trial, and draw our conclusions as far as We 

 can from it. 



It would be easy to calculate the probability that the number of bulls in any one 

 compartment should exceed a given number ; but in order to make this investigation 

 applicable to the general proljlem of periodicities we must ])roceed in a different 

 manner. If the compartments are numbered — it does not matter in which order — 

 and a curve be drawn in the usual manner representing the connection between 

 the compartments and the number of balls in each, we may, by Fourier's analysis, 

 express the result by means of periodic functions. The amplitude of each period 



can be shown on the average to be \^nn It is often more convenient to take 



■m 



the square of the amplitude — call it the intensity — as a test, and we may then say 

 that the ' expectancy ' of the intensity is injm'-. The probability that the intensity 

 of any period should be /.; times its average or expectancy is e"''. We may apply 

 this lesult to test the reality of a number of coincidences in periods which have 

 been suspected. A lunar effect on earthquakes is in itself not improbable, as we 

 may imagine the final catastrophe to be started by some tidal deformation of the 

 earth's crust. The occurrence of over 7,000 earthquakes in Japan has been very 

 carefully tabulated by Prof. Knott according to lunar hours, who found the Fourier 

 coefficient for the lunar day and its three first sub-multiples to be 10-U, 17-9, 10.9, 

 .'!-n7; the expectancy on the hypothesis of chance distribution for these coefficients 

 I find to be 10-3, 15-7, lO'Ci, 502. The comparison of their numbers disproves the 

 supposed connection ; on the other hand, the investigations of Mr. Davison on 

 solar influence have led to a result much in favour of such influence, the ampli- 

 tude found being in one series of observations equal to Kve times, and in the other 

 to fifteen times the expectancy. The i)robability that so large an amplitude is due 

 to accident in the first case is one in ^00 millions, and in the second the probability 

 of chance coincidence would be represented by a fraction, which would contain a 

 number of over seventy in the denominator. We may therefore take it to be 

 established that the frequency of earthquakes depends on the time of year, being 

 greater in winter thin in summer. With not quite the same amount of certainty, 

 but still with considerable probability, it has also been shown that earthquake 

 shocks show a preference for the hours between 9 a m. and noon. 



A great advantage of the scientific treatment of periodical occurrences lies in 

 the fact that we may determine a jjriori how many events it is necessary to take 

 into account in order to prove an effect of given magnitude. Let tis agree, for 

 instance, that we are satisfied with a probability of a million to one as giving us 

 reasonable security against a chance coincidence. Let there be a periodic effect of 

 such a nature that the ratio of the occurrence at thf time of maximum to that at 

 the time of minimum shall on the average be as 1 + X to 1 —X, then the number of 

 observations to establish such an effect is given by the equation ti =- 200/\^ 



If there are 2 per cent, more occurrences at the time of maximum than at 



1. L2 



