Page:Somerville Mechanism of the heavens.djvu/420

330 $wgUseTeX = true; If s be the latitude of Jupiter, by article 436 center|s=qsinv—pcosv; hence h = jg. 6in i^ — }p. cod o, and substituting for Sp, Sg, J, = - _m«7i_ fdP ^^^^ _ „ + n)- ^' . 8iii(x-i,+n)} bji' — 2n 1^7 J7 (177) which is the only sensible inequality in the latitude of Jupiter m tiiii approximation. The latitude of Jupiter above the pihnitiv^ orbil of Saturn is « = — 7 sin (© — n) whence — J«:= S7 sin (r — 11) — 7411 cos (« — II) and a comparison of the two values of it, gives 57' = J — cos A* + — sin X bn' - 2n  dy dy * ^n/— m^,an {dP ^ .x dP . ^1 yjn'=— —; { — . cos X — Bin X>. 5n' - 2n I dy dy J These are the variations occasioned by the action of Saturn in the mutual inclination of the two orbits, and in the ascending node of their common intersection; but Jupiter produces a corresponding effect in these two quantities, and if it be expressed by iy"j fW^ then tlic whole variations will be 47 =: «y' + «7", >n a JD' + iW; but by article in'.an m'.an or, substituting for n and n the whole variations in the two ele- ments in question are im! .an m sTa^-^ m'Va' j dP' dP m'. an m ^ a + m' Vo' i dP'. dP 'yjn = +5n'- 2n- ^^7^7 • I-57- ««» ^ " rff cos X}. 577. The corresponding periodic inequalities in the latitude and elements of the orbit of Saturn are