1890.] Harmonic Analysis of Tidal Observations. 299 



by the total number of entries gives the required results. But for 

 N, L, and similarly for the diurnal tides K 1? O, P, the grouping and 

 summations have to be broken into a number of subordinate periods, 

 which are to be operated on to form S and S* w . The multiplication 

 by the eighteen mean cosines and sines is best deferred to a late stage 

 in the computation. 



Thus, for example, for N and L, the quarter-lunar-anomalistic 

 periods, i, ii, iii, &c., are treated independently, and we find 

 (lt_3id) : t : (2 Dd 4 th reversed) for each. There are thus eighteen 

 cosine numbers and eighteen sine numbers for each of i, ii, iii, &c. 



We next form the snms two and two, i + ii, iii + iv, <fec. ; next find 

 the differences (i+ii) (iii +iv), (y+vi) (iii-f-iv), &c.; add the 

 differences together ; then multiply by the eighteen cosines or sines 



of 2, 7|, &c., and finally multiply by 4,/ n +i\f m _iy and 8O find 



. 

 sin 



We next go through exactly the same process, but beginning with 

 ii instead of i, and so find S*"7i . 



The same process applies, mutatis mutandis? for finding S and 



sin' 



There are two cases which merit attention in particular. The 

 sorting of heights in quarter-lunar-anomalistic periods, according to- 

 values of V m , serves, in the first instance, for the evaluation of N and 

 L, but it serves, secondly, to evaluate M 2 , for we then simply neglect 

 the subdivision into quarter periods and treat the whole as one series, 

 but stop at the end of a semi-lunation. 



The sorting of heights in quarter-lunar periods, according to the 

 values of \V m , also serves several purposes. 



We first find from it S and S*"7i |F m , and secondly, by merely 



bill 



counting the entries in each group for each quarter period, instead of 

 adding up the heights, we arrive at S and S* ff cos F m . (It may be 

 noted in passing that what is wanted, according to preceding analysis, 



is the sum of C ? s (|F TO V m ), so that there will be a change of sign in 

 sin 



the sine sum to get the desired result.) 



But, besides these, S and S^ c ? 8 QF m + F m ) can be obtained with 



8111 



sufficient accuracy from the same sorting. 



The angles F m were sorted in four times eighteen groups, for each 

 quarter-lunar period. If each angle were multiplied by three, the 

 eighteen entries of the 1 st quadrant would be converted into three 

 groups of six, lying in three quadrants, viz., I st , II nd , III rd ; the 



Y 2 



