ON THE HABMONIC ANALYSIS OF TIDAL OBSERVATIONS. 43 



It might, perhaps, be useful to evaluate the diurnal component on the 

 M sheet, for if it does not come out small it is certain that the amount 

 of observations analysed is not sufficient to give satisfactory results. 



In the article the harmonic analysis is arranged according to a rule 

 devised by General Strachey, which is less laborious than that usually 

 employed, and which is sufficiently accurate for the purpose. 



§ 2. On the Notation employed. 



It will be convenient to collect together the definitions of the prin- 

 cipal symbols employed in this paper. 



The mean semi-range and angle of lagging of each of the harmonic 

 constituent tides have, in the Tidal Report for 1883, been denoted gene- 

 rically by H, c ; but when several of the H's and (.'s occur in the same 

 algebraic expression it is necessary to distinguish between them. The 

 tides to which we shall refer are Mg, Sj, N, L, T, R, 0, P, and K2, Kj ; 

 the H and k for the first eight of these will be distinguished by writing 

 the suffix letters ^, 5, „, &c., e.g., H„, k^ for the Mj tide. "With regard 

 to the K tides, we may put H", k" , and H', k' . 



Again, the factors of augmentation f (functions of longitude of moon's 

 node), as applicable to the several tides, will be denoted thus : — for M2, 

 N, L, simply f ; for K2, Kj, f", f respectively ; for O, f^. 



The K2, K, tides take their origin jointly from the moon and sun, and it 

 will be necessary in computing the tide-table to separate the lunar from the 

 solar portion of K2. Now, the ratio of the lunar to the solar tide-generat- 

 ing force is such that •683H" is the lunar portion and '31711" is the solar 

 portion of H". 



In the Report of 1885 a slightly different notation was employed for 

 the H's and /c's, but it is easy to see how the results of that Report are to 

 be transformed into the present notation. 



As in the Report of 1883, we write t, h, s for local mean solar hour- 

 angle, sun's and moon's mean longitude, and r, i,, v', 2i'" for functions of 

 the longitude of moon's node depending on the intersection of the equator 

 with the lunar orbit ; also y — //, t], a, tn are the hourly increments of t, h, s 

 and longitude of moon's perigee, and e, e, the eccentricities of lunar and 

 solar orbits. 



Let p, p, denote the cubes of the ratios of the moon's and sun's paral- 

 laxes to their mean parallaxes ; 0, l^ the moon's and sun's declinations ; 

 p' the value of p at a time tan ((>■„, — ^■'D)li'^ — '^)> orlOS'^'S tan (/.•„ — v^) 

 earlier than i ; 0' the moon's declination at a time tan (//' — '<^m)/2c, or 

 o2'^-2 tan (k"—k^) earlier than t. 



Let P, P^, P' be the cube roots of p, p^, p'. 



Let A, A, be declinations such that cos^A, cos-A^ are respectively the 

 mean values of cos^?, cos-c^ : obviously A has a small inequality with the 

 longitude of the moon's node. 



Let e be an auxiliary angle defined by 



Hn sin K„ — H-i sin Ki 



tan e = fj ^Ft * 



M^cos K^ — Mjcos /Cj 



Lastly, let 4/, ;/', be the moon's and sun's local hour angles. 



