HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. lll 
analysis has been carried out, in degrees per m.s. day, the values of cos Ii 
and sin /¢ have been computed for =0, 1,2 .. . 364, so that there are 
730 values for each of the five tides. These 730 values have then been 
multiplied by the 365 ¢éh’s corresponding to each value of ¢, and the 
summations gave =ch cos lt and S ¢h sin It, the numerical results being 
the left-hand sides of one pair of the ten final equations explained in 
§ 10. Now, it appears that this labour may be largely abridged, without 
any substantial loss of accuracy. 
The plan proposed by Professor Adams is that of equivalent multi- 
pliers. The values of cos /t may be divided into eleven groups, according 
as they fall nearest to1:0, :9,°8,°7....°2,°1, 0. Then, as all the values 
of ch are to be multiplied by some value of cos J, and that value of cos It 
must fall into one of these groups, we collect together all the values of 
6h which belong tc one of these groups, sum them, and multiply the sum 
by the corresponding multiplier, 1-0, -9, -8, &c., as the case may be. 
Since there are as many values of cos It which are negative as positive, 
we must change the sign of half of the ¢h’s. This changing of sign may 
be effected mechanically as follows :—In the spaces for entry of the oh’s, 
those ¢ h’s whose sign is to be unchanged are to be entered on the left side 
of the space if positive, and to the right if negative ; when the sign is to 
be altered this order of entry is to be reversed. Thus in the column 
corresponding to each multiplier we shall have two sub-columns, on the 
left all the ¢%’s which, when the signs are appropriately altered, are +, 
and on the right those which are —. The sub-columns are to be 
separately summed, and their difference gives the total of the column, 
which is to be multiplied by the multiplier appropriate to the column. 
The treatment for the formation of 8h sin It is precisely similar. 
The annexed form [Schedule R] is designed for entry for deter- 
mination of 2c h cos («—n)t. 
The entries of ¢h are to be made continuously in the marked squares 
from left to right, ard back again from right to left. The numbers in 
the squares, which in the computation forms are to be printed small and 
put in the corner, indicate the days of observation. The rows are 
arranged in sets of four corresponding to each complete period of 2(¢—n). 
In the middle pair for each period the + values of 5h are to be written 
on the right, and in the rest on the left. The word ‘ change’ opposite 
half the rows is to show the computer that he is to change the mode of 
entry. Hach column, excepting that for zero, is to be summed at the 
foot of the page, and multiplied by the multiplier corresponding to its 
column. A pair of forms is required for each tide of long period; they 
are very easily prepared from the existing forms, in which the values of 
the multipliers are already computed. 
