68 Prof. Sylvester's Astronomical Prolusions. 



dt a 2 f c 1 • 



so that j- = -=-< 1 cos m V, which proves the point m ques- 

 tion*. 



The force/ for this case has been given by Newton in the 

 third section of the Principia ) it can be obtained instanta- 

 neously from the equation 



n a dv 2 p^ + cfi—c 2 



v 4 = qf cos i = 



so that 



or 



2 dp %P 



dv 2 —4ap 



jiJ _c__. f - 9l 



„2\2> J ~ 



Calling p' the remainder of the chord K of which p is a part, 



so that /varies as 



1 



pW 



as given in the Principia, and of course, if the force-centre is 



at the extremity of a diameter, / varies -3, which is the case in 



which our two reciprocal foci come together. When one of 

 them is at the centre, the other goes off to infinity, and the 



Lb Lb r 



actual amount of force exerted by it, ^, or ~ • -j- 5 , becomes zero 



when ~ is finite ; so that this case returns to that of a single 



force at the centre of the circle. If we wished to find the 

 general law of the respective forces//' at the two reciprocal foci 

 suitable to produce motion in the circle we might proceed as fol- 

 lows : — Calling i, i' the angles between the radii vectores drawn to 

 these points from any point in the circle and the radius at that 

 point, and writing 



V=tfdr.f, V'=2j«fr'./', 



* Hence follows the statical proposition that the force which tending to 

 any centre retains a point in a circular orbit may be resolved into two 

 forces tending to two fixed centres, each varying as the inverse fifth power 

 of the distance : this proposition will be generalized subsequently in the 

 text. 



