192 ON THE CONSTANT TERM IN THE [25 



These relations are established by means of the same principle which 

 was employed to prove the theorem above mentioned, viz. that B = Q and 

 (7-0. 



They were, however, arrived at much later, namely on August 14, 1877. 



ANALYSIS. 



Let x, y, z denote the rectangular coordinates of an imaginary Moon 

 at any time t, the plane of xy being that of the ecliptic, and the axis of 

 x the origin of longitudes. 



Also let x', y' be the rectangular coordinates of the Sun, t j its radius 

 vector, and pf its mass. 



Then if we neglect the terms which involve the Sun's parallax, the 

 equations of motion are 



d'x u.x u!x ?>u!x' 



= 



d'z LLZ u!z 



I ~ I _ f\ 

 ~TI^ ' -- 7" T ---- /< V" 



at- r r 



Now let x l} y u z l be the rectangular coordinates, and r^ the radius 

 vector, of another imaginary Moon at the same time t as before, so that the 

 same equations of motion hold good, and p., //, x', y', and r' are unaltered. 



'Xi fj.x t ux 

 Hence ~ + ' + 



l 



^ , A _ n 

 3 H " = 



Multiply the first set of equations by x u y v z t respectively, and subtract 

 their sum from the sum of the similar equations in x u y v z 1 multiplied by 

 x, y, z respectively. 



