Periodicities when the Periods are unknown. 385 



36 



simple harmonic series whose periods are of the form - — , that is, it 



N 



contains all the harmonics of 36 days ; and in the third pair of 

 brackets of the expression, we have arranged those terms of the 

 simple harmonic series whose periods are greater than the period 

 36 days. Suppose in the fifth difference and sum series, or in the last 

 horizontal line of our form, we have 36M values where M is some 

 large integral number. Suppose we have another table ruled with 

 more than M horizontal and 36 vertical lines. In the first horizontal 

 line let us enter 36 numerical values from the commencement of the 

 series in order, in the second horizontal line the next 36 numerical 

 values, and so on, in the Mth horizontal line the last 36 numerical 

 values. Let the sum and mean of each vertical column be taken in 

 the (m + l)th and (m + 2)th horizontal lines respectively; then 

 these means are the subject of our present investigation. 



37. The general term of the simple harmonic series in the first 

 horizontal line is — 



^sin (U + P6); 



the general term of the same simple harmonic series in the same 

 vertical column of the second horizontal line is — 



wsin (U+P6 + 36&) ; 



the general term of the same simple harmonic series in the same 

 vertical column of the third horizontal line is— 



wain {U + P6 + 2 (366)}; 



the general term of the same simple harmonic series in the same 

 vertical column of the Mth horizontal line is — 



^sin {U + P6 + (M-1)(366)}; 

 thus the mean of this vertical column will be — 



smM-^ 



;in j(U + P6)+?^36& j 

 then the general series may be written — 



u sin | + M ~ X 36^ + Fb j 



JV1 sm 



2 



M ~2~ 



sin - 



. '66b 

 M sm — — 

 2 



Thus the last horizontal line of means in our new form, which is 

 composed of 36 values, will be of the form — 



2d2 



