MR. LUBBOCK ON THE THEORY OF THE MOON. 139 



As the preceding results do not quite agree with those of M. Plana, I shall en- 

 deavour to show how they may be obtained from the same equations which he em- 

 ploys. 



d^^ 



i? = - -^ 1 1 + 3 cos (2 X - 2 \) j 



In order to integrate the preceding differential equation, let 



— = |M» { 1 -|- Tq + ^1 cos (2 X — 2 w X) -f- e cos (c X — w) + e Tg cos (2 X — 2 mX — cX -f tsr) } 



[0] [1] [2] (3] 



The letters Tq, r^, &c., being now used in a somewhat different sense to heretofore, 

 having now reference to the expression for — in terms of tlie true longitude. 



Neglecting e^^, 



-s 



'^ + 5^^0+3^) = 0- 



Substituting in this equation for h its elliptic value which is allowable, rQ= — —, 

 alsor3=^m, 



(1) =l;|l+;;^2 + ?|x3m2e2^-?|^+3m8e2| 



r* ^' r I , 3 ^« 1059 2 2 1 4 /^ 



= a 1 1 + ^ + m2 -f- -^ m2 e2 j 



f 1 2 2 2 739 2 2 1 

 = a I 1 — e2 - y m2 ~ Y^ m2 e2| 



= a -j 1 — e2 — -gj w2 e2 I as before. 



t2 



dx 



/l2 



