196 ON FORCES OF ATTRACTION 



seems impossible to avoid the conclusion that an ellipse, 

 and the same ellipse, must be described; because one of 



the forces alone acting from S, as -j, would give the ellipse 

 passing through S and 2' ; and the other force alone acting 

 from S, as -, would give the ellipse passing through the 



same points S and 2' ; and the initial velocity * and angle 

 of projection would prevent any difference in the length of 

 the conjugate axis ; and in the middle point answering to 

 the centre C, the equality of r and q and of P a and Pa' 

 would make the diagonal PC coincide with the conjugate 

 axis. But a further combination of forces may be sup- 

 posed in this case ; two forces acting towards the points 



T (1 



S and S' and in the proportion of r and q. or -- and . 



m m 



How will this addition affect the locus of P? It should 

 seem, for a reason similar to that before given, that the 

 curve would remain the same; for the two new forces 



T fl 



- and , acting in r or q or PS and PS' respectively, 

 m m 



their resultant must, if there were none other acting, pass 

 through the middle point C, between S and S' ; and as we 

 know that a force acting from that point, and in proportion 

 to the distance from that point, causes the body to move in 

 an ellipse whose centre is that point, and r -f- q being con- 

 stant, the ellipse must have the same axis and coincide 

 with the ellipse produced by the combination of the forces 



TO m 



and -5-. 

 r z q* 



7. This had appeared to be a necessary consequence of 

 the conditions stated, but not as at all proving the ve- 

 locity to be the same in the ellipse, when described by 



* The condition of Legcnclre (mentioned in page 193), that I 2 = V 2 + if 

 ia supposed to hold ; for otherwise the centrifugal force would not be 

 sufficient to balance the centripetal. 



