On Lunar and Solar Periodicities of Earthquakes. 459 



certain namber of successive figures eliminates to some extent the 

 shorter periods, and therefore emphasises the longer ones. But this 

 elimination is done in a much better and more complete manner by 

 Fourier's analysis, and when it is therefore intended to submit the 

 figures to that process of calculation, the smoothing down of the 

 original numbers, represents a waste of arithmetical work, uncom- 

 pensated, as far as I can see, by any advantage. 



To judge of Mr. Knott's results it is necessary therefore to trace 

 the effect of the process of calculation employed by him. The 

 problem to be solved for that purpose may be stated thus : — 



A number, p, of figures, y u y 2 . . . . y p , is expressed in terms of the 

 periodic series (1). A second series is then formed by taking the 

 sum of m successive numbers, the first two members of the series 

 being y } + y 2 + .... y m and y 2 -f y 3 + .... y m+1 . The original series is 

 supposed to repeat itself, so that the last member of the derived 

 series becomes yp + yi + yz + . . . . + 2/«t-i« It is required to find the 

 relation between the coefficients of the Fourier expansion for the two 

 series. 



It will be sufficient to find the solution of this problem for the case 

 that the original series y u y 2 represent equidistant ordinates of the 

 curve y = cos kx, so that 



y x = cos kx u y z — cos (ksJi + S), y 3 — cos (*s& 1 + 2£) .... 



y m == cos (ic^i+(m — 1) c). 



The solution of this case really includes the general one, for the 

 original series y is supposed to be represented by a number of terms, 

 each of which is simply periodic, and to each of which the result of 

 the special case may be separately applied. 



The first term of the derived serie3 may be obtained by a well 

 known process, for 



, , , sin ^mS ( , m — 1 - A 



y = y 1 + y 2 + ....*.= -^jj- COS (^+-— « j • 



The subsequent terms are obtained simply by altering the value 

 of Xi. 



The derived series, y', y", &c, represents therefore the equidistant 

 ordinates of the curve 



sin|ra<$ / (w-}-l)£\ 



V = — — Vr cos kx + — • 



u sin V 2 / 



It is a curve having the same period as the original one, but 

 having an amplitude reduced in the ratio sin|m£/sin \ h. Applying 

 this result to the separate terms of the series (1) and remembering 

 that 8 represents the difference in phase between two successive 



