THE ORBIT OF URANUS. 49 



Pert urhaf ions of Lon(jitude. 



The perturbations of the longitude are now to be computed by formulae (24). 

 To do this in the most simple way we remark that tlie numbers given on page -42, 

 dR 



dv 



we put 



under the heading ^ -, are those represented in formula (42) by w^ and i\. If 





nt = t' 

 equation (24) may be put into the form 



but we have from (42) 



C ^^ 7. ^ /"*'l' • XT "''j* aA 



I -^-dt = 2, 1 ?7„siniv 1' cosiv). 



If now we represent the numerical values of cos ^bp, already found, by 



2(p,siniV^+p,cosiV), 



and if we substitute these expressions in the above value of " , the latter will 



at' 



become 



^ = rr'^ |(v, - 2p,) sin N-{- (v, - 2p,) cos N^ 

 where we put for brevity 



v^ = — 3Lv,. 



The numerical expression for r~- is given on page 40, and by multiplying the 

 quantities within brackets by this expression, after tlie manner explained on pages 



40 and 41, we form the terms of — -. Multiplying each of these terms by its 



do 



corresponding value of v, changing cos to sin and sin to cos, Ave have the coefficients 



in the expressions for ^v given on page 50. 



As previously mentioned, before commencing the above computation, I had 

 computed all the perturbations of Uranus by the method of " perturbations of the 

 elements," using the formulae developed in my Investigation of the Orbit of Nep- 

 tune. The two results are here placed side by side, for the purpose of comparison. 

 The discrepancies in the various coefficients, expressed in thousandths of a second, 

 are shown in the sixth and seventh columns. 



It will be seen that the largest discrepancies, and indeed the only ones (with a 

 single exception) exceeding one-tenth of a second, occur in the coefficients of the 

 terms 2j'—l aug 3(/' — I. Here the errors are almost certainly in tlie computation 

 from perturbations of the elements. Owing to the long period of the term S</ — I 

 they would not become sensible in the course of any one century. 



7 April, 1873, 



