MR. LUBBOCK ON THE THEORY OF THE MOON. 135 



d7? + -^ - 2F7? ^^^^ (2 X - 2X, - ^X + - sin (-X - .)} 



~ 8^ {^ ^^s (^ - ^ - ^?^ + + y cos (X - X^ + ^X - 



+ 9 cos (3 X — X^ - ^X + v) + 6 cos (3 X - 3 X^ + ^ X - v) I 



The simplest method of substituting for X^ in terms of X seems to me to be by first 

 obtaining expressions for cos X^ sin \, cos 2 X^, sin 2 X^, in terms of n^ t. Having obtained 

 these expressions, they may be reduced to terms of X by Lagrange's theorem*; but 

 when the higher powers of m are neglected, it is sufficient to write m X instead of n^ t. 



sin2X^= (1 — 4e/)sin2w^f — 2emi{2n,t — I') + 2 e^ sin (2 tz^^ + ^ 

 -{■^ e^^ixi {2 n^t- 2y +-^e,2sin(27i^^ + 2Q + &c. 



cos 2 X^ = (1 - 4 e^) cos 2 nj — 2 e^ cos {2n,t— Q + 2 e, cos {2n,t -\- ^) 



+ ^e^2cos(2w/- 2y + ^e2cos(2w^# + 2y + &c. 



Great facility results in the following substitutions in consequence of the coefficients 

 being alike in the corresponding arguments of the expressions sin X^, cos X^ ; sin 2 X^, 

 cos 2 X^, &c. ; so 



sin (2 X — 2 \ — g X) = (1 —4 e^) {sin (2 X — gX) cos 2 m X — cos (2X — ^ X) sin 2 m X} 

 — 2e^ {sin (2 X -- ^X) cos (2 m X — c^m X) — cos (2 X — gX) sin (2 w X — c^mX)} 

 + 2e, {sin(2X — ^X) cos (2mX + c^mX) — cos (2X — ^X) sin (2mX + c wX)} 



* By Lagrange's theorem, if 



•^ 2du 2.3dM2 ^ 



So «,%"'=! + l-e,^[\ +^e,^)+3e,[l+l.e,'^)cosl 



m\ = n,t+2me(^l - ^j sing + A me^ (l - ii e^j sm2g 



+ — ^ wi e* sin ? + -— — m e* sin 4 P + &c. 

 IZ 96 



Hence evidently 



+ •|-e/'^(l +y e^2jcos(2c^wA) + ^ e^3cos (3c^mA) + ^ Vcos(4c,»iA) 

 + terms multiplied by m, c^ may be considered as equal to unity. 



