AND SEMI-ANNUAL SEISMIC PERIODS. 
nil 
ordinate in the curve so obtained is reduced in the ratio 2 : tt. It is also further reduced 
bj the method of grouping the earthquakes in monthly intervals. For, instead of 
using the actual ordinate for cos 6, we use the average ordinate over an interval tt/S, 
that is 
- cos 6 dO, or — sin — cos 9. 
73'J0-V12 I^ 
Now, the value of 12 sin tt/tt is '9886. Hence the difference between each 
reduced six-monthly mean and unity must be multiplied by the factor ^7r/’9886 or 
1’589. The same remark applies to the case of taking six-half-monthly means for the 
semi-annual period. In like manner, in investigating the latter period by means of 
monthly intervals and taking three-monthly means, the corresponding augmenting 
factor is 1'645. 
7, Comj'yarison of the Methods of Investigating the Semi-Annual Period .—The 
method here employed for investigating the semi-annual period seems to me to possess 
several advantages. (1) It is less laborious to apply than Dr. Knott’s method ; (2) it ■ 
avoids the interpolation of arithmetic means in the series of three-monthly and six- 
monthly meaus ; and (3) it eliminates, or diminishes more completely, the effect of 
shorter periods, if any such should exist. The latter point requires a brief 
examination. 
Taking Dr, Knott’s method first, we have to evaluate the ratio 
2rt„ 
niT 
Sill -p - Sin q- 
IT 
Ctji 
a.-, 
for different values of n. If n = 3, the value of this ratio is '943 ; if n — 4, it is 
zero; if n = 5, it is ’566 ; if n = 6, it is Hence, compared with that of the six- 
monthly period, the amplitude of the four-monthly period is slightly decreased, that 
of the period of one-fifth of a year is diminished by nearly one-half, while that of the 
two-monthly period is diminished by two-thirds. 
In the method adopted in this paper we have to evaluate the ratio 
2tt ,i sin4 a7r 
2fq/7r ’ fq 
for different values of n. If n = 2, its value is zero ; if n = 3, it is Hence, the 
three-monthly period is eliminated, and, compared with that of the six-monthly 
period, the amplitude of the two-monthly period is diminished by two-thirds. More¬ 
over, as will appear from § 10, both the four-monthly period and the period whose 
