42 



the origin being focus, h c and e being the arbitrary constants intro- 

 duced by integration. 



3. The application of this method to the case of a planet acted on 

 by a disturbing force is worthy of particular notice, as it expresses 

 the variations of the elements of the orbit with great facility, in the 

 following manner : — 



If U be the symbol of the disturbing force, we have 



fe)=D M .U (1.) 



dt v 



&.=E.R&R.\J + \J (2.) 



dt h* 



These two equations determine with great facility all the elements 

 of the orbit. For y is a direction unit perpendicular to the plane 

 of the orbit (i. e. it is the symbol of the pole of the orbit), and there- 

 fore it defines completely the position of the plane of the orbit. Also 

 s is a direction unit in the plane of the orbit at right angles to the 

 axis major, and therefore it determines the position of the axis major ; 

 in fact the direction unit of the axis major is Dy.e. The letters h 

 and e have their usual signification. 



To find h and y separately from (1.), suppose that we obtain by 

 integration of (1.) 



hy=W ; 



W 



then A 2 =AW.W ; and h being thus found, we have y= — -. The 



h 



same observation applies to (2.). 



4. The expression for the parallax of the planet is 



^ + A/3y^(^/3A/3.U + u). 



These instances suffice to show the nature of the proposed sym- 

 bolical method. 



