462 



Prof. Schuster. 



assumption that the earthquakes are altogether independent of each 

 other and take place at random. Any regularity would be contrary 

 to this assumption and might affect the expectancy. Thus, for 

 instance, it is known that earthquakes take place in groups, a large 

 earthquake being generally followed by some minor shocks. 



Mr. Knott states that the i4 obvious " aftershocks were left out of 

 account in his calculation; but there may be aftershocks which are 

 not obvious, and it seems quite likely that every earthquake is 

 followed by a period during which another is more likely to happen 

 than at other times. If there is such a tendency it is easy to see 

 that our calculated numbers for the expectancy will be too low. 

 Take the case, for instance, that all earthquakes happen in groups of 

 two, or, what comes to the same tbing, let each earthquake in the 

 investigation of § 2 count as two. The quantities I have called p> 

 will not be affected by this change, but the total number of earth- 

 quakes being doubled, the expectancy calculated according to our 

 formula is now reduced in the proportion of ^2 to 1, and would 

 therefore be too small in that ratio. It seems to me to be probable 

 that the discrepancy between Mr. Kcott's coefficients and the calcu- 

 lated expectancy is explained in this way. Apart from this possible 

 explanation it would not be safe to draw any certain conclusions 

 from any instance in which the calculated amplitude has double the 

 value of the expectancy, for expression (9) shows that the ampli- 

 tudes will turn out to be even greater than that on the average in 

 one case out of every twenty-three. The matter to be explained is 

 not that any one of Mr. Knott's coefficients is, roughly speaking, 

 twice as great as the expectancy, but that all coefficients show this 

 tendency towards higher values in not very different proportions. 



6. It is interesting to discuss, from the point of view of this paper, 

 the periodicities of earthquakes which apparently depend, directly or 

 indirectly, on the position of the sun. We owe to Mr. Davison a 

 very complete discussion of the annual period.* The method 

 employed by him (in determining the amplitudes of the periodic 

 terms), though neither direct nor very accurate, is sufficient for our 

 purpose. Taking as an example the record of 5879 earthquakes in 

 the northern hemisphere, given by Mr. R. Mallet, the results of this 

 paper show that if they were distributed indiscriminately over the 

 whole year the expectancy for the amplitude would be V 7r/5879, or 

 0*023, while Mr. Davison, in § 18 of the paper quoted, gives O'll for 

 the amplitude. Similarly, the discussion of 8133 earthquakes, for 

 which the expectancy is 0'020, yields the number 0'29 for the ampli- 

 tude. Here, then, we have the amplitude in one case equal to five 

 times, and in the other equal to fourteen times, the expectancy. 



The probability of the accidental nature of so large an amplitude 

 * 'Phil. Trans.,' A, vol. 184, p. 1107 (1893). 



