Prof. Sylvester's Astronomical Prolusions. 67 



If we call M the absolute force of the sun, it will readily be 

 seen that we must have 



a 



■ a 



where c, y are the distances of the two new centres of force from 

 the centre of the planetary orbit, and satisfy the equation 



2ac 



which gives 



c 2_^ c + ^ = 0, 

 e 



a — b a-\-b 



b representing the semi-minor axis, c being equal to 



a(\-V\-e>) ,_,._„ Vl-e* 



e 



c-ae = a v x { </l- e <2-l\ 



and is always negative, so that the interior centre of force 

 always lies between the centre of the orbit and the sun ; when e 

 is small it lies about midway between these two points, but 

 nearer to the latter than the former : ex, gr., if we were to sup- 



3 i u i 3a a(1— \/\— e *\ a , . , 

 pose e= -, we should have ae= , -^ L=s -, which 



Sa e 



differs not very much from — • 



It is perhaps remarkable — at all events I am not aware whether 

 any one has remarked, that the motion of the eccentric follower 

 of a planet may also be brought about by a single force placed 

 at the sun itself, attracting according to the law which is con- 

 sistent with the body describing a circle. This is immediately 

 apparent ; for if we call S the position of the centre of force, 

 C the centre of the circle, c the distance of S from C, a 

 the radius of the circle, P any point in it, calling i the angle 

 SPC, u the angle PCS, we have 



h h h 



v = 



p jocosz a — ccosu 

 P 2 



