Page:Somerville Mechanism of the heavens.djvu/422

Since i^ and ^^' represent the great inequalities of Jupiter and Saturn, their corrected mean motions are nt + 8f , and n7 + 8f ' ; and, by the substitution of these in the preceding equation, it becomes (Sf) = - 6m' ff . an}dt* . 2Q . sin {bn't - 2nt + 5«' - 26 - ^ + 5ir' - 28f } (181) (it) being the great inequality of Jupiter when the mean motions are corrected. In order to abridge, let bn't — 2nt + 5e' — 2« = X, then sin (X - /8 + 5if' - 2J0 =3 sin (X - i8) cos (5Sf - 2Jf) + cos (X - fi) sin (5if ' - 2Jf). But 5i{f' ^ 2i{; is BO small, that it may be taken for its sine, and: unity for its cosine ; and as quantities of the order of the square of the disturbing forces are alone to be retained, sin (X — /8) may be omitted ; hence sin (X - /8 + 58f' - 2J?) = {5Jr - 2Jf} cos(X - fi) ; m tfa or, as if ' = - , ,— • if therefore Bm(x-.ff+5Sr-2?f)=J^^ ^^■f2m-^| ^ j^..^^,^^) but the integral of equation (180) is vv," 6m' . an* .2.0 . ^ o ^f == — TTl — -^-^ . sm (X - ^), consequently sin (X - /3 + 5if ' — 2jf) = (3m' . an? . 2 .Q)« f 5mVV+ 2m' V"?l . ,^^ When this quantity is substituted in equation (181), instead of the sine, its integral tv^^ (3m'.an«.20)« f 5m^+2mV?l . o/». .* o i . «. . S^^^X ^^^= '■2(5;^ny-l ^.V^ )' "" 2(5n'^2n<4.5.'.2cH3) is the variation in the mean motion of Jupiter, and on account of the