THE OR13IT OF NEPTUNE. 



17 



= net ( 

 = ue( 2 



The values of JV W are as follows : 



N< = X K 



= - a,' + 2 A o 



2 a/ a <y 



a *f- o' - 2 o 

 = _^'+3^ 2o 

 = 2A a -co' 

 = 3 A' ^ 2<y 



_a,'_2o) 



3o>' 

 a' 



vf 



(19) 



^ 2/1' 2/1 



2 



2 



6> 



zz 3 



(20) 



A'+ 2^ 2 o' 

 4/1' /L 3o>' 



From these values of N the corresponding values of v are derived, remembering 

 that 



ri 



~ ' 



i' and i being the coefficients of /I' and 2. respectively in the value of N. 



The check on the correctness of the preceding values of V, R, and B may now 



n 



be applied by developing v in powers of , and retaining only the first term ; 



/& 



that is, by putting v z= ^, v 2 =. 0. Making these substitutions, all the values of 



% 



V, R, and B will be found to vanish. In other words, [j? will be the lowest power 

 of /j. which will enter into the values of V, R, or B, as we have already shown from 

 a priori considerations. 



For convenience, we shall give the values of V, R, and B developed according to 

 the powers of n, the ratio of the mean motion's, a form similar to that in which 

 the lunar inequalities are developed in the theory of the moon. Putting 



we have 



s y 



May, 1865. 



