in Mechanics and Physical AJlronomy.

to a series (P'xcos. pz-\- Q':xcos. qz-\- R'xcotrz, Scc.) of co-sines of multiples ot the arch z, joined to small, given, coeffi-cients P', Q', R', &c. and let the force n, acting in the per-pendicular direction, be also supposed, as the distance directly,drawn into a series (Pxsin.M -bQxsin.yzr-j-Rxsin.re * &c.)of sines of multiples of the same arch, joined to small, given,coefficients P, Q, R, &c. According to these assumptions, by

substituting bx 1— wV — x P'cosi/z-j-Q^cos. qz &c. and—— x Psin. te-l-Qsin .qz &c. for their equals A and IT, our

1— w 1

ITi t

two equations, 2 = I -\- 2 fluent , and — w =

JL x 2 — A x i— w\~ z — n x ~x i— toP 3 , will here become

2 — 1+2 fluent P z fm.pz + Qjz sin. qz Scc. X i—W>l + , andf- 4 - w — -h into 2 —b — P'co i.pz -j-Q^cos. qz &c. x i —w! 3

zz 2

— - xP sm./z-j-OJin. qz &c. x i— toP 4 .

Now, the orbit being supposed nearly circular, we may, inorder to a first approximation, neglect w in both the factorsi—w' -3 and i—- to) 4 , as being very small in respect of unity;by which means 2 will become = i +2 flu. pz sin.

sin. qz &c. — i-\-d —— x cos. pz —x cos. qz &c. (feep. 82.) where d, representing the necessary correction to the flu-

2P

ent, must be taken = ~J + “ 6cc. so that 2 may be —i,when 2;=o. This value of 2 being now substituted in the se-cond equation ™ 4 -m = 4 x 2— b —P'cos. pz —Q^_cos. qz &c.

(where ^-xP sin. p£-{-QJjn. qz &c. on account of the small-

* This assumption is not the less general by the multiples of z being taken the famehere as in the value of A ; because , if any multiple of z, in the one values entersmt into the other , it is but supposing the corresponding coefficient in this lajl, to va-nish or become equal to nothing.

U 2 ness

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