136 MR. LUBBOCK ON THE THEORY OF THE MOON. 



+ -^e/{sin {2X — gX)cosi2niX-2c^mX) - cos (2X — g-X) sin (2mX — 2c^mX)} 



13 

 + -^ e/ { sin (2 X — ^ X) COS (2 m X + 2 c^ m X) - COS (2 X — ^ X) cos (2 m X + 2 c^ w X) } 



= (1 — 4 e^2) sin (2 X — 2 m X - ^ X) 



— 2 e^ sin (2 X — 2 m X + c^ m X ~ ^ X) + 2 e^ sin (2 X — 2 m X ~ c^m X — g X) 



+^e^2sin(2X — 2mX + 2c^mX — ^X) + ^e^2sin(2X — 2mX-2c^wX--^X) 



^=:m2|l +ye2^3e,cosc^mX+ -g- e/ cos2 c^mXj 



{l+2e2— 4e cos c X + 5 ^^ cos 2 c X} 



2-^3 = m2|— +-je/ + 3e2-6e,coscX + -2 e^cosc,mX+-2-e2cos2cX 



27 ") 



— 9e^cos (eX — c^mX) — 9 ee^cos(eX + c^wX) + "J ^/cos2 c^mX > 



All the terms which I have verified in the expression for the latitude in terms of tlie 

 true longitude agree with those given by M. Plana. 

 IfX = nt -{■ K, then by Taylor's theorem, 



^d 5\ . /d^ 5\ a^ . /d^ s> 



^=i^-> + 0^+{m^ + {^h'+^- 



(a) being the quantity arising from the substitution of 7i t for X, in the expression for s 

 in terms of X. In this manner I found the same terms as those given by M. Plana, 



15 . 15 



except — o^ m e2 y sin (2 r — 2 | + >?) instead of — gr m e2 y sin (2 r — 2 | + ^)j and 



3 1 



— g- m e gj y sin (2 r 4- I + 1^ — r) instead of -g- m e e y sin (2 r + I + I, — n)- 



I next obtained ^2^ in order to procure the terms in the longitude depending on y2. 

 The quantity e in my notation does not accord with that quantity in the work of 



M. Plana, but with e ( 1 — -^ ) ; so that, in order to arrive at the same figures in 



some of the terms multiplied by y2, this circumstance must be attended to. 

 I find 



^70 = I - 16^' - 16^' - 32^ - m- ~ ^ m2 ^ -^2 j _ = _ 



39 

 instead of •— ^ m^, according to M. Plana ; and I find 



37 2 

 32 '^ 



f35 , 7 S5 7 7 1 7 



^85= -|-8-w*+ 8-^-16 '^- 8- '^- 16^1= -T 



m 



7 

 instead of -g- m, according to M. Plana. 



