302 REPORT— 1876. 



Hence if »j denote the mean angular velocity of the sun's radius vector, y 

 the angular velocity of the earth's rotation, and © the mean sun's right 

 ascension (or, which is the same, the sun's " mean longitude ") at the instant 

 of the first transit of Y after the vernal equinox of the year, we have 



=?il+2ec»(,-.)} approximately 



=^ I '\.+2ecosf^x+Q—'^) \ approximately. 



Henceforward ^ must denote the whole angle turned through by the earth 

 from the instant of the first transit of Y across the meridian of Greenwich 

 after a time when the sun's longitude was zero. 



Hence, integrating, 



0=© + ^X+2«8in(^^^ + O-wj. 



Now, if A denote any angle, 

 8in(-2^+A)= i 



/'_ 20 -?^X + ^)-'i^ cos /'-20-?^X + ^) sin (-X + O-^) approximately: 



and therefore as 



p»=P'ri + 3«cosnx-wj1 approximately. . . . (XL) 



we have (XII.) (see page 301). 



Going back now to (VI.), (VII.), and (VIII.), and attending to (X.), use 

 (XI.) in the first term of (VI.) and the last term of (VII.) ; neglect the 



variation of parallax and put ^=— x+ O ^^ the small terms of (VI.), (VII.), 



and (VIII.) ; use (XII.) in the first terms of (VII.) and (VIII.), giving to A the 



respective values X —/ and^ + 2(x— ^); and collect as in (IX.) : we fiud 



(XIIL), (page 301). 



To obtain the corresponding expression for the moon's equilibrium tide, 

 substitute in the preceding, h' for h, M for S, P' for P, I for w, e' for e, o- for 

 T), }> — y for O, -ro-' — I' for zs, f+f for /, and g+y for g : M denoting tlio 

 moon's mass, v the right ascension of the ascending node of the moon's orbit 

 on the earth's equator, ]) the mean moon's right ascension at the time of that 

 transit of T across the meridian of Greenwich from which x (as stated above) 

 is reckoned, ■cr' the longitude of the moon's perigee, o- the mean angular 



smi 



