The formula for 1 which in pp. 295, &c. (a2-2aa' cos. w +a2)} was deduced for the conveniently determining of the differential dA dB coefficients Indeed, as it is plain, it is requisite for expanding the fourth &c. will now serve another purpose. term. By means of it, then, and of the expanded forms of Chap. XVII. we may express R in a series of cosines of the mean motions, &c. dR dR and thence, immediately, the values of > de' da &c. But in this research of the value of R it is a very important point to determine whether it contains any constant quantities. If there should be such, involving either the inclination, or the nodes, or the perihelion, &c. then, (see p. 420.) some of the elements would necessarily have secular variations. In order to determine these constant parts in the value of R we must extend its development beyond the forms of pp. 276, 277, 279, and include terms involving e2, e, eé: but, it is not proposed to include terms involving higher powers or products: for, it is a supposition, in this as indeed it has been in all preceding enquiries, that e2, é2, ee', sin?. ? are very minute quantities. We will examine the terms of the expression of R in their order; and, it will be convenient to premise, that we shall be principally guided in this examination by looking after terms the factors of which contain the cosines of similar arcs: for, it is plain, (see Trig. form [7], p. 27.) that one of the terms of cos. (m t + a) cos. (m t + b), when expanded must be constant and equal to ¦ cos. (a−b). Make p1, and the sixth term of the value of cos. (v — v), (see p. 275.) is constant and equal a but the first term of, when expanded, must be ; there is, therefore, on this account, a constant term introduced equal to é 1—e cos. (nt + e − π) + 2 é'. cos. (n't + e − x') - 2 e é'. cos. (nt + e − π) . cos. (n't + e − π') Now the third and fifth terms of the value of cos. (v v) (see p. 275.), are - e cos. (n' t + e − π), - e'. cos. (nt + e − π), which combined, respectively, with the third and second terms of thirdly, one of the cosines, resulting from developing the fourth ee'. cos. (n't which combined with the first term of the value of cos. (vv), (see p. 275.) produces Now, (see ll. 4, 14, 20.) these four constant cosines destroy each other consequently, at least up to éé, &c. there are no constant terms in we may, therefore, combine the value of cos. (v' m'r 22 with the value of - (see p. 410.) v) (p. 275.) and deduce some terms involving the cosines of constant arcs (π/ T): the coefficients, however, of such terms will, at the least, involve ee. But quantities (see p. 409.) involving sin2. ed are not to be taken account of. 2 = (if we take account merely of the constant parts) 2 dД e'2 d2 A e2 dA A+ a. +a. 2 + a2. da 2 d a2 4 d2 A é'2 'da 4 + &c. the constant part of P' will combine with the constant part of cos. (vv), which, see p. 275, is the sixth term of its value, and equal eé. cos. ( - ), and form a constant quantity. #), The constant part of P' (see p. 276.) is similar to the above value of P (1. 13.) and equal, d B e2 but, for reasons already stated (see p. 409. 1. 17.) we need only reserve the first term B, which multiplied into m'ee cos. (π' — π) produces In deducing the other constant quantities of P' cos. (v' — v) we must proceed on the principle laid down in p. 409. 1. 22, &c. Now the third term [-e. cos. (n' t +ε' — π)] of cos. (v bined with the third term da v) com Δά of P' (see p. 276.) when in such term ▲ a' is expressed by its first term, namely, -a' é cos. U, and e. cos. (nt+e), the fifth term in the value of cos. (— v) (see p. 275.) combined, similarly to the above combination, with Aa, produces d B da Lastly, cos. (n't—nt + e' €), the first term of cos. (v - v), combined with d2 B da.da P'), when, instead of A a, Aa', their first terms Aa. A a' (the sixth term of the value of These three last terms (11. 12, 16, 24.) then being multiplied by-m': the sum of the constant parts of m' P' cos. (v'—v) is There are no constant terms, within the prescribed limits (see p. 409.) to be derived from P'cos. (2v′ −2v), P'. cos. (3v' — 3v), &c. cos. (2 v′ — 2 v) (see p. 275.) contains no constant quantity: and the first constant quantity produced on the principle of p. 409. 1. 22, &c. is by the combination of its first term, namely, d2 B cos. (2 n't-2n1+2-2 e) with .Aa. A a', when for Aa, da.da' : substituted but then the coefficient of the resulting term would involve e2 e'2. +(B'+ &c.) cos. (n't nt + e − e) + &c. + &c. the only constant term then, of which it is necessary to take account, is have If F, then, be used to designate the constant part of R, we |