APPENDIX. 423 



d y d 2 x) = 2 (ydx-xdy + cd y ) 3 , or ^~ = 



r2 (y d x x d y + c d ?/) 3 



The difficulty follows -- wherever that proportion en 

 ters into the investigation. Thus in the problems con 

 nected with different centres, when it is found that 



forces varying as - 2 and 2 , being combined with forces 



varying as the distance directly, or as r and q } give 

 an elliptic orbit, the resultant of the latter forces passes 

 through the centre, and the locus of that resultant is 

 the opposite semi-ellipse, and so of a circle. But when 



the proportion is -- and , (also if the force towards 



each centre is as the radius vector to the other centre,) the 

 resultant passes through innumerable points to an opposite 

 curve, sometimes of a different kind, although each result 

 ant differing in its direction from all the others, and in the 

 case of the circle, from the diameter, is equal to the one 

 passing through the middle point of the line joining the 

 two centres. In this case, therefore, there is no combined 



action of the forces - 2 and - 2 , or ^ and - 5 or of their seve 

 ral resultants, with the resultant of - and -, as there is in 

 the case of - and -, but the several forces act wholly in 



m 



the direction of the radii vectores severally. 



It evidently appears to be a more simple and natural 

 combination that the two sets of forces should dimmish 



E E 4 



