36 THE ORBIT OF NEPTUNE. 



ACTION OF JUPITER (Continued). 



6v = (! log r = 



P. K. N. P. K. N. 



+ 0".030 sin ( /' + 3 I 2 TT) 

 .011 sin ( I' + J 2 TT) 

 + .004 sin ( 2 I' 2 w) 



+ 0.10 0".005 sin (2 I ir 1 TT) 

 + .028 sin (2 V ir 1 ) 



<?/? = 



+ 0".5G4 sin (I T) 

 + .039 sin (I' r) 



By comparing the different authorities for the coefficients, it will be seen that 

 while our present results agree very well with those of Professor Peirce, the 

 agreement with Professor Kowalski is in many cases very far from being satis- 

 factory. It will be observed that the latter differ most in the case of those terms 

 whose coefficients depend on the action of the disturbing planets on the Sun, and 

 we have also seen that these terms are ordinarily developed as small differences 

 of very large quantities. They are, therefore, the terms, into which errors would 

 most easily creep. 



The terms enclosed in parentheses are not of great importance, because they 

 are for a long period sensibly confounded with the elliptic elements. Notwith- 

 standing that one of these terms amounts to more than half a degree, and others 

 to several minutes, the effect of the whole of them could scarcely be discovered 

 from all the observations hitherto made on Neptune. 



17. For the purpose of tabulating and computing an ephemeris, it is expedient 

 to change the form of the perturbations by Uranus. Consider any two terms in 

 which the coefficients of 7 are equal, but of opposite signs : 



&v =PI sin { si' i A u } -f- Pi sin { si' -\-iA o> } 

 where 



The terms may then be put in the form 



{ (p. 2 p^) sin o sin iA + (PI + p\) cos a cos i A } sin si' 

 { (p z pj) cos sin i A (p 2 -f p^) sin &> cos i A } cos si' 



So that we may put 



$v = to. + P,. t sin I' + P, t cos Z' + P s . 2 sin 2 I + P c . 2 cos 2 1' 

 8 log r 5 log r -f- R tl sin Z' -f- P^ cos Z' 



where &v, P, and R are functions only of A, and may be tabulated as such. 



18. For Jupiter and Saturn, if we neglect those terms of which the coefficients 

 are less than 0".03, it will be more convenient to tabulate the perturbations 

 directly. This course we shall adopt, except with reference to those perturb- 

 ations which depend on the mean longitude of Neptune alone, and do not contain 

 the mean longitude of the disturbing planets. These have been omitted by both 



