460 



Prof. Schuster. 



'terms of the original series, it is seen that the coefficients a 2 and b q 



are both reduced in the ratio sin V^l. / s in 7r ^. This result lias to be 



p I p 



•divided by m to fit it to the case that the derived series is not 

 formed by taking the sum of m successive numbers, but by taking 

 their mean value. 



The effect of smoothing down the irregularities of the observed 

 numbers has therefore the effect of reducing the ^th coefficients of 

 ^the Fourier series in the ratio 



. 7rmq / . tto 

 sm — i- / m sm — i . 

 pi p 



In the case under discussion, m = 5, p — 25, so that the ampli- 

 tudes of the first four coefficients of the series are reduced respec- 

 tively in the ratios 



0-938, 0-765, 0-517, 0-244. 



The number of earthquakes taken into account was 7427, so that 

 the expectancy for the amplitude of any one of the coefficients, irre- 

 spective of the smoothing process, would be 



</irlim — 0-0206. 



If this number is multiplied by the above fractions, and then 

 by 1000 in order to make the units agree with those of Mr. Knott's 

 paper, we obtain the numbers placed in the first row of Table I ; 

 the second row gives by comparison the coefficients actually found by 

 Mr. Knott. 



Table I. 



Coefficients. 



Ci. 



c 2 . 



c 3 . 



C 4 . 



Expectancy for the coefficients by the 



19-3 



15-7 



10-6 



5-02 



theory of probability 













10-3 



17 -9 



10-9 



3*97 



The numbers in this table do not support Mr. Knott's contention, 

 but seem to me rather to be a striking confirmation of the theory of 

 probability. It must be remembered that the " expectancy " only 

 gives the average value of a great many cases, the individuals of 

 which may differ considerably from that average. Thus it may be 

 calculated with the help of expression (9) that in about one case out 

 of every four the coefficient Ci would come out still smaller than the 

 number found by Mr. Knott, while the coefficient C 3 would be larger 



