the Coefficients of the Lunar Inequalities. 491 



equalities, and especially in an elaborate treatise published by 

 Sir J. Lubbock some years since in the Philosophical Transac- 

 tions. The process of calculation, however, by either method is 

 sufficiently laborious and complicated to render any mode by 

 which the difficulty would be lessened a desideratum. To en- 

 deavour in some degree to supply such a mode is the object of 

 the following paper, in which it will be observed the calculation 

 proceeds directly from the fundamental equations of motion, thus 

 avoiding the process of integration. 



Dublin, October 1866. 



] . The three equations of motion referred to polar coordinates 

 are as follow : — 



d 2 p dv 2 _, _ 



_£_,_ + P=0 , 

 dpfo d% T _ 



£m+s=o. 



For the purposes of calculation the following forms are con- 

 venient : — 



dv 2 d 2 p 



Pd?-i?- p=0 > w 



[Vg] +3 T P =0, (2) 



d f n adv 

 di 



jM+S=0 (3) 



2. In the lunar theory — P, 2Tp, and S may be thus ex- 

 pressed : 



-P=-^(l~|^+^ 4 )+im / -^+|m i ^cos2(t;--«; / ) 



p 2 p 2 



-f fm^cos (v — vd+^mfacosSiv—v,) ; 



8 < p * " 8 - p 



,2 



2Tp = 3m l ^sm2(v — v l ) + jm^sin (v— v y ) + 1 ^^ l ~ i smS(v—v l ) ; 



Pi Pi 



