LE VERRIER ON THE PERTURBATIONS OF PLANETS. 347 



(2 \2l-3 ^ ^ 



— =) sin(2i-3)-...sin- 



x(^)"" 



sin (2i — 2)--.. . sin 2 



2 J 



:2(i- 1)3 V/-1. 



It is from the care which we had in taking the second and 

 third of the equations of the system (19.) that the first members 

 of the equation (25.), which have ah'eady been calculated, are 

 again found here as factors. If we have regai'd to these condi- 

 tions and to the notation (24.), we shall be able to put for short- 

 ness. 



sin (2i — 2) 



/r',_, = [^-l],-[i],- 



F,_, = [^-l]3-[^],- 



sin (2« — 2)- 



(27.) 



These expressions reduce the preceding equations to the fol- 

 lowing : 



Their form is the same as that of the equations (25.), and we 

 shall deduce from them in a similar manner. 





sin(^-])(2^ + l)- 

 sin (i— 1)« 



cos(i-l)(2i.fl)^ 

 sin (i— 1)« 





sin (z-l)(2i-l)- 



sin (i— 1)« 

 cos(i-l)(2«-l)^ I 



. . (29.) 



sin («—!)« 



