THE ORBIT OF NEPTUNE. 9 



From the first equation and tlie relation between the mean distance and mean 

 motion, we obtain 



^— =z 3 m'ra'^ I'll sin N. 

 at 



These equations are entirely rigorous, provided that we regard the elements in 

 the second member as variable. But they can be integrated only by successive 

 approximations. In a first approximation the elements are regarded as constant 

 Equations similar to (1) for the elements of all the planets whose action is taken 

 into account being integrated in this way, the resultiiog values may be substituted 

 in the second members of (1), and a new integration be performed. 



In the case of Neptune, however, the variations of the elements are so slow 

 that a single approximation will be amply sufficient for a period of several cen- 

 turies, provided that we adojit suitable values of the elements in the second 

 members; that is, if we add such constants to the integrals that the latter shall 



be very small for the present time. Putting vz:z — 



i'n'-\- irb 



we shall have, on the supposition that the elements as they enter into the second 

 member are constant, 



log a' — mvA cos N-\- a'o, d zz. mvE cos iV+ e'o, (2) 



I' zz mvL sin N-\- n'J, + e'o, n' = mvWs>\n iV+ Tt'o, 



A, L, E, and W being given by the equations 



A— 2, i'h, 



' ' da ^ clef' 



!!= — h{j' cot 4^' + i'g'), (3) 



Tr= cot 'J/' -=--• 



«e' 



a'o, n'o e', e'o, and ti'q are arbitrary constants, dependent on the position and velo- 

 city of the planet at a given epoch, a^ and Hq are, however, dependent on each 

 other. 



For the perturbations of the true longitude in orbit, and the logarithm of the 

 radius vector, we shall have, omitting accents, 



Sv = 8l 

 + mv{eL — eW—E}sm{N— f) — i e'mv {eL — eW—oE} sin [N—f) 

 -\-7nv{eL — eW—E} sin (N-^ /) —le^mv{eL — eW-]-3E}sm{N-\-/) 

 -}-iemv{eL — eW—E}sin{N—2/) —etc. 



+ f emv {eL — e W-{- E } sin {N+ 2/) (4) 



+ ¥ ehnv {eL — e W— E } sin {N— 3/) . 



+ ¥ ehnv {eL — eW+E} sin (iV+ 3/) 

 + W e'mv {eL — e W— E } sin (V— 4 /) 

 + etc. 



2 May, 1865. 



