300 REPORT— 1876. 



Hence by (I.) and (II.), 

 D = 2arr{cos(x— J'— /)cosa + sin(j( — V— /)sma}sin5co8S, . . (IV.) 



S=-5-G{cos2(x— v — (7)cos2a + sin2()^-i^— 5f)sm2a}cos'^^. . . . (V.) 



Now S and a are the two legs of a right-angled triangle of which ia the 

 hypotenuse and I the angle opposite to S. Hence, by spherical trigonometry, 



sin 2^ sin Isin^, 

 cos a cos ^=cos <p, 

 sin a cos S= sin cos I. 

 Hence sin^S= ^ sin'^I(l — cos 2^), 



and so L=|arK(— f +sin'I— sin'I cos 2^) (VI.) 



Next for the diurnal tide : 



sin S cos S C08 a = sin I sin <p cos 0=-^ sin I sin 2^, 

 and sin h cos S sin a=sin I sin^^ cos 1=^ sin 1(1 — cos 2^) cos I ; 



and using these in (IV.) we find (VII.), page 301. 



Similarly, towards reducing for the semidiurnal, 



cos 2a cos^^=(2 cos^a — 1) co8^S = 2 cos°0 — (1 — sin^I sin^^) 



= cos 20(1 - 1 sin^I) + i sin'I = cos 2^(1 + 1 cos'I) + i sia'I, 



and sin 2a cos^3=2 sin a cos h cos a cos h =co8 I sin 2(p. 



Using these in (V.) we find (VIII.), page 301. 



The sum of these three expressions (VI.), (VII.), and (VIII.), 



7i = L + D + S, (IX.) 



would be the required complete simple harmonic expansion, if t were constant, 

 and if <p increased simply in proportion to the time. 



To complete the process we must, by aid of physical astronomy, express r 

 and (p in terms of y. 



For the case of the sun, the only deviation from uniform circular motion 

 which produces sensible influence on the tides is the elliptic inequality ; for 

 the case of the moon we must take into account also the perturbations called 

 evection and variation. 



Por the case of the sun we have 



r=||p^ l = w;v = 0; (X.) 



if E denote the earth's mass, S the sun's mass, p his parallax at any time, 

 and u) the obliquity of the ecliptic. Let P denote the mean parallax, ^ the 

 longitude of the perihelion, and e the eccentricity of the orbit. As f now 

 denotes the sun's longitude, we have by the polar equation of the ellipse with 

 one focus as pole, 



^=P{l + ecos((i» — ro)} ; 



and by Kepler's first law —^ -^ is constant. 



J- /L 



