TRANSACTIONS OF SECTION A. 



333 



with the earliest given ahove. (When we group in 5x7 years we have the groups 

 1700-1734, 1735-1769, 1770-1804, 1805-1839, 1840-1874, and the years 1875-1899 

 do not suffice for another complete group.) 



Earthquakes recorded in 1840-1874. 



Sum 



X sin 8 + 124 + 109 + 55 - 59 - 137 - 119=- 27 



xcosfl +143 + 99 - 25 - 114 - 122 - 31 + 95 = + 45 



It is clear that the —27 and +45, variations in a mean value of 967/7, represent 

 essentially smaller quantities than the —19 and —14 previously, which are variations 

 on a mean value of 167/7. We should express the quantities as percentages to get a 

 fair comparison ; and this suggests multiplying by 100 and dividing by the mean 

 values 167/7 and 967/7. But the division seven is compensated in another way. 

 If we had a term A sin 9, then on multiplying the seven values by sin 6 and sum- 

 ming them we should get 7/2 (or if there were n terms we should get w/2). The seven 

 thus cancels out and leaves finally 



200/total number. 

 Thus in the second case we get 



a= -27x200/967 b = +45x200/967 tan- , a/&=329° 



= -5-6 =-\ — 9-4 a 2 +& 2 =120 



The phases tan _1 a/6 are only needed when we get a suspected periodicity, 

 first instance we only want the values of « 2 +6 2 , which are as below. 



Table I.— Values of a 2 + 6 2 . 



In the 



The difference in the number of groups is due to the fact that in 200 years we can only 

 get two groups of 5 x 20 years ; but we can get four groups of 5 x 10 years. 



Now it is clear that the accidental variations in the early records are a much larger 

 percentage of the whole than later. This is only natural. For instance, if the records 

 for a whole year were lost, this would reduce the value of any minimum falling near it. 



The correspondence of the groups is very roughly indicated by the broken line 

 dividing the whole series into two : the mean value of a 2 +6 2 for the earlier half is 300, 

 and for the later half 140. 



It seems also pretty clear that there is nothing likely to be a real periodicity in 

 this part of the periodogram. For such a real periodicity the value of a 2 +6 2 should be 

 at least five times the average. Let us first consider possible periodicities of exactly 



