T II E OK KIT OF URANUS. 49 



of Longitude. 



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



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



"3 /? 



under tin- heading ' , are those represented in formula (42) by v, and v r If 



we put 



nt = ( 



equation ('21) may be put into the form 



a 



but we have from (42) 



If now we represent the numerical values of cos 4-fy>, already found, by 



I, (p, sin N-\- p c cos N), 



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



at' 



become 



~ = r,-2 {(v, - 2p.) sin N+ (v. - 2p c ) cos N\, [ 



where we put for brevity 



v, = Mvv e , 



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

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



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



etc 



corresponding value of r, changing cos to sin and sin to cos, we 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 

 dements," using the formula? 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 2/ I aug 3g' I. Here the errors are almost certainly in the computation 

 from perturbations of the elements. Owing to the long period of the term 3</ I 

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



7 April. 1878. 



