18 THE ORBIT OF NEPTUNE. 



We shall also put 



V rV 

 V 

 yi -a*-ff 



_R _cR 



^-a 3 -^' 



B cB 



c being a constant, equal to unity if we neglect the change of mean distance pro- 

 duced by the action of other planets. We then have 



! sin N, 



8 log r = mac^R^ cos N, (21) 



sin N. 



Substituting the above developments for the vs in V, R, and B, we have 



Vf = (1 u z i-e 2 ) ( 1 i? 6 fj. s 19 ft 4 ) 

 + ^(1 2(j. 2 30p 8 ) 



VV =>(- |- f;U 2 - \lt?) 



i^ 2 + fl 3 + f|M 4 ) 

 1 2 + V 3 + 25^) 

 3^ 2 27 ^ 3 135 ^ 4 ) (22) 



_ ^ ( _ y _ y ^ _ 2| 



25 



_l_4 2 24 



=^ ( I 6^ 2 3(V 116 ^ 4 ) (23) 



zre 2 ( f T V^ 2 - i 1 ^ 3 ) 



