THEORBITOFURANUS. 25 



Making this substitution, and putting also 



* +*+/+i = — '• 

 these equations become 



dyi man \ t^i , ,- , s i } ■ »r 



--, = ^ 1" \ ^ + (* + j) <7^* \ sill N 



at «! cos T^ cos §/ I 2cr ) 



(27) 



dk m'aa cos iy Ch -.r 



= ?^ cos ^. 



<7< 2(/i cos ^ da 



To pass to the general rotations dp and dq, let us represent by 0,, Oo, etc., the lon- 

 gitudes of the ascending nodes of the several orbits of the disturbing planets on 

 that of the disturbed planet. AVe shall then liave 



dl "^"'^'^f ^'"'^•- dt 



^- = VeosO.-^^; + 2sm0,-;;. 



(28) 



These equations completely define the instantaneous motion of the orbital plane. 

 They cannot, however, be rigorously integrated in their present form because p 

 and q as integrals have no completely defined signification. To do this it is neces- 

 sary to express the differential rotations dp, dq, etc., in terms of tlie difterentiuls 

 of any elements we may select to define the position of the orbital plane, and then 

 to integrate the equations thus formed. But, for the purpose of constructing tables 

 of the planets we may consider j<, q, etc., to represent small rotations of the planes 

 of which the powers and products may be neglected, and the integration is then 

 quite simple. 



Pertvrhations of the second order dep>cnding on the motion of the orbital qdaiies. 



R being a function of the five quantities of ;■, r, v, v', and y, the motion of the 

 orbital planes introduces terms of the second order by changing the values of v, v', 

 and y. These terms we have hitherto neglected. To investigate them let us refer 

 the rotations of both planes as given by (28) to the node of the disturbing on the 

 disturbed planet as the principal axis. If we represent by dyj, dk, d/;', and dk' the 

 rotations corresponding to this axis, and designate by the subscript 1, the quantities 

 which refer to tlie disturbing planet whose action we are considering, and by 2, 3, 

 etc., the other planets, the equations (28) will be replaced by these 



the summation commencing with i = 2. 



By formula? of the same kind we are to find the differential rotations dy;' and 

 dk' of the orbit of the disturbing planet, produced by the action of all the planets. 



4 April, 1S73. 



