ON THE 



GENERAL INTEGRALS OF PLANETARY MOTION. 



§ 1. Introduction. 



If we examine what has been done by geometers towards developmg the co- 

 ordinates of the planets in terms of the time, we shall see that the most general 

 expressions yet found are those for the development of the secular variations of 

 the elements in a periodic form. It is well known that if Ave neglect quantities 

 of the third order with respect to the eccentricities and inclinations, the integra- 

 tion of the equations which give the secular variations of those elements, and of 

 the longitudes of the perihelia and of the nodes, leads to the conclusion that the 

 general expressions of those elements in terms of the time are of the form 



e sin n^^i Ni sin {g^t -\- (3i) 

 1 



e cos 7t ^ i.i Ni cos (git -\- (3i) (1) 



1 



^ sin = 2< Mi sin (hit + y^) 



<|) cos = ii Mi cos (hit -j- yi) 

 1 



n being the number of planets, iVJ, Mi, g^, and 7^ being functions of the eccentrici- 

 ties at a given epoch and of the mean distances, while (3i and y^ are angles depend- 

 ing also on the positions of the perihelia and nodes at a given epoch. It is to be 

 remarked that one of the values of Zi; is zero, the corresponding quantities M and y 

 depending on the position of the plane of reference. 



The numerical values of these constants for the solar system have been found by 

 several geometers. The latest and most complete determinations are those of 

 Le Verrier and of Stockwell.-^ 



When we consider the terms commonly called periodic, that is, those which 

 depend on the mean longitudes of the planets, we shall find that their determina- 

 tion depends on the integration of differentials of the form 



m'h 1°^ (w + ii +yw +y7t + m + m), 



where we put 



m! the mass of the disturbing planet. 



' Smithsonian Contributions to Knowledge, No. 232. Vol. XVIII. 



1 October, 1874. 1 



