THE ORBIT OF URANUS. 



CHAPTER IV. 

 TERMS OF THE SECOND ORDER PRODUCED BY THE ACTION OF SATURN. 



Preliminary Investigation of the Orlit of Saturn. 



FOR the accurate determination of the perturbations of a planet it is essential 

 that the functions of the time which are substituted for the co-ordinates of each 

 planet in the expression of the disturbing forces should approximately represent 

 the true places of the planet. The difference between the true place and that 

 implicitly assumed in the investigation should be so small and of such a character 

 that, when multiplied by the mass of the disturbing planet, and by the factors 

 introduced by the process of integration, the result shall be insensible. If one of 

 these factors is so large as to make a- perturbation of an order of magnitude approxi- 

 mating that of the inequality which gives rise to it, it will represent an inequality 

 of very long period in the elements, which, though apparently sensible, may be 

 neglected for a great length of time. 



The perturbations hitherto found have been computed on the hypothesis that 

 the disturbing action of Saturn on Uranus is the same as if both planets moved in 

 the elliptic orbits corresponding to the adopted elements. We have given formulae 

 for the computation of the corrected perturbations when, to the co-ordinates of the 

 two planets corresponding to the adopted ellipse, we add corrections represented 

 by &v, &v', 5p, etc. These corrections are now to be taken of such magnitude that 

 when thus added they shall very nearly represent the actual motions of the planets. 



Generally, it is considered sufficient to take for these corrections the perturba- 

 tions of the first order. But this presupposes that the elliptic elements are nearly 

 correct, which does not hold true in the case of the old elements of the outer 

 planets. Bouvard's Tables of Saturn, the elements of which have been adopted, 

 are subject to recurring errors amounting to 30" or more. Moreover, when we 

 substitute the new and more accurate perturbations for the old and imperfect ones 

 adopted in the tables, the chances are that the errors will be increased. Desiring 

 that the theory shall be as far as possible free from doubt, we begin with a pre- 

 liminary investigation of the orbit of Saturn, the design of which will be to give 

 the co-ordinates of that body in terms of the time with sufficient certainty and 

 accuracy to serve for computing the perturbations both of Jupiter and Uranus. 

 As usual, the first step in this investigation will be the determinations of the per- 

 turbations of the planet. 



