TO SEVERAL CENTRES. 205 



tance 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 resultant 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 - and -, or and - or of 



r <f r 5 (f 



their several resultants, with the resultant of and , as 



j^ ;;:.-. r q 



T (1 



there is in the case of and , but the several forces act 

 m m 



wholly in the direction of the radii vectores^ severally. 



It evidently appears to be a more simple and natural com- 

 bination that the two sets of forces should diminish with the 



distance increasing, as in - and r combined with and 



r 2 (f . r 



, than that one set should decrease and another increase 

 q i 



with the distance, as in and - with r and a. in which 



r- (f 



case there must even be an extinction of force at one point, 

 where (taking the sum of the forces instead of their resultant) 



m m r -4- q q a m* 



- = -. or r is as in the equation r a H 



r <f m q* 



r 2 = m 2 . Of course the value of q would be the same ; and 

 the resultant (more accurately taken to measure the increase 

 of the force) would at one value give the two sets of forces as 

 counterbalanced. 



