142 Cambridge Philosophical Society. 



bolical method, that the symbol of the force acting on the planet is 



\dt° / r dt 



where la is the angular "velocity of the planet, and to' that of the 

 plane of the orbit about the radius vector. The expressions for the 

 three component forces along r, perpendicular to r, and perpendicular 

 to the plane of the orbit, are the coefficients of a /3 7 in this expres- 

 sion. 



2. The equation of motion of the planet, when the force is the 

 attraction of a fixed centre varying as the inverse square of the di- 

 stance, is 



d'^u _ jua 



It is curious that this equation is immediately integrable, the in- 

 tegral being the two equations 



r h 



The latter equation is the symbolical equation of a conic section, 

 the origin being focus, h c and s 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 



^M:=VU,\J (1.) 



dt 



^)=if!:/3Aj8.U + U. ..... (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 01 the plane of the orbit. Also 

 8 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.g. 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.) 



then A2=:AW.W ; and h being thus found, we have y= — . The 

 same observation applies to (2.). 



