THE ORBIT OF URANUS. 25 



;ikin^ this substitution, and putting also 



these equations become 



dy man \ ih .. .. , ) . 



- '= - 4-( -4-i)a?i \ sm N 



dt t/ t cos 4, cos $y I la t 



(27) 

 <// I/I'IIH cos ly 6h xr 



. = ^- *' COS.Y. 



</< 2^ cos 4> da 



'!'( pa-s to tli. general rotations </j> and dq, let us represent by 0,, fc etc., the lon- 

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

 that of the disturbed planet. We shall then have 



dk ( 



r 



(28) 



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

 Tln-y cannot, however, be rigorously integrated in their present form because p 

 and </ as integrals have no completely defined signification. To do this it is ncces- 

 to express the differential rotations dp, dq, etc., in terms, of the differentials 

 of any rlnm-nts 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 p, q, etc., to represent small rotations of the planes 

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

 quite simple. 



/' -turbatioru o//7te second order depending on the motion o/ tlve orbital planes. 



/i 1 being a function of the five quantities of r, 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 dri,dk, dy, and dM the 

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

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

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



*- +S ft- W *' -S sin (*_ *. 



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. 1873. 



