430 APPENDIX. 



attraction as the mass is the same from S and S , or that 

 the mass of the body in S and in S is the same ; then 

 it 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 2 , would give the 

 ellipse passing through ^ and ^ ; and the other force alone 

 acting from S, as ^, would give the ellipse passing through 



the same points ^ and 2 ; and tie 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# 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 



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 



M . SY 



and , acting in r or q or PS and PS 7 respectively, 



T 



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 + q being con 

 stant, the ellipse must have the same axis and coincide 



* The condition of Legendre (mentioned in page 427), that I 2 = V 2 + v- 

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

 cient to balance the centripetal. 



