AND SEMI-ANNUAL SEISMIC PERIODS. 
1109 
and 
S 
= 1 — 
72 - = 
v/2 
a, v/2 
TT 
COS ( 2 ^ -|- oto) “ 1 “ . . . 
^nir — v/2 sin ^nir) cos {n9 + ««) + ...; 
where in T all terms are absent which involve multiples of 4^, in S all those 
involving multiples of 26, while, in the third expression, the terms involving 6 and 
all multiples of 4^ disappear. 
If, then, we take 27r to represent a period of one year, T represents the result of 
taking three-monthly means, and the effect of this is to eliminate the^ three-monthly 
period, if there be one, and all periods which a.re fractions of three months, and also 
to diminish the amplitudes of all other periods relatively to that of the annual 
period. In like manner, S represents the result of taking six-monthly means, the 
effect being to eliminate the six-monthly period, if there be one, and all periods 
which are fractions of six months, and also to diminish the amplitudes of all other 
periods relatively to that of the annual period. The result of the third operation is to 
eliminate the annual period, and to give special prominence to the six-monthly and 
four-monthly periods, if such exist. 
Now, if we have a record of all the earthquakes felt during a given interval 
(containing an exact number of years) over a definite area, and if we group together 
all the earthquakes felt during each month of the year, we may regard the numbers 
so obtained as representing the average number for each month felt on the middle day 
of that month. The mean of the numbers for the months of December, January, and 
February, will be the three-monthly mean corresponding to the middle of January; 
and the mean of the numbers for the months of November, December, 'January, 
February, March and April, will be the six-monthly mean corresponding to the end of 
January. If three-monthly and six-monthly means be calculated in tlrs manner for 
each month, and if the results be plotted in the usual way, the first of the curves so 
obtained will show no trace of a three-monthly period, and the second none of a six- 
monthly period. To eliminate the annual period, arithmetic means are interpolated 
between each pair of numbers in the two series, the means of the first series giving 
the three-monthly means corresponding to the end of each month, and those of the 
second series the six-monthly means corresponding to the middle of each month. 
Multiplying each number of the first series by Ij^2, or ’707 , and subtracting it 
from the corresponding number of the second series, the curve obtained by j^lotting 
the resulting difierences will show no trace of an annual period, and will give 
special prominence to the six-monthly and four-monthly periods, if such exist. 
Lastly, if each of the twenty-four difierences be divided by the average of all, the 
curves so obtained for different intervals and for different areas of observation will be 
more directly comparable with one another. 
