134 MR. LUBBOCK ON THE THEORY OF THE MOON. 



I find 



M. Plana has -gj m^, and for the numerical value of the coefficient converted into 



sexagesimal seconds "-607. M. Damoiseau has "*90 ; I obtain "'615. 

 I find 



>^49 = {2^49 + ^31 + ^9 + ^49 + ^31 + T "^AwT^^) = ^ ' 



M. Plana has — -gj m. 



I find 

 I find 



T *»Q1 77 



>^59 = 2 ^59 X 4^ = -yg m, M. Plana has — 3^ w^; 



ho = {2 rgo + Kfio) w—f^) = --^m. 



{2 - tim) 



These discordances will appear very trifling, considering the nature of the calcula- 

 tions ; and it is by no means impossible, after all, that M. Plana may be right, and 

 that the mistake may be with me, notwithstanding all the pains I have used. 



Before the terms in the longitude can be arrived at which depend on y'^, it is ne- 

 cessary to obtain the expression for the tangent of the latitude s : this may be done 

 by means of the equations 



d^2 jx 2 _,iniZ 3 m^zrr cos (x' — ^/) ^ ■ 



d7^+73- + 7TH —^ — I 



* = 7- + 275 - &c. 



tan ~ s •= s — s "^ ~5 ~ ^^' 



It is, however, more convenient in the determination of s, to adhere to the method 

 of Clairaut, that is, to the method adopted by M. Plana, notwithstanding the diffi- 

 culties which occur in that method, and to which I have before alluded. 



The following is the differential equation employed. 



Substituting in this equation in the terms multiplied by m^, for s, r sin {g 7^ — v), 



ds 

 and for j^, — r cos {g7^ — v), neglecting the square of the disturbing force and the 



cube of s, I obtain 



