66 Prof. Sylvester's Astronomical Prolusions. 



value of a to the radius, the period becomes —\ «(a 2 -fc 2 ), which 



if 



of course is the same as — a(a 2 + y q ). 



If we now suppose the two absolute forces fi, fi', and 3 the dis- 

 tance between them, to be given, the problem of determining the 

 magnitude and position of the orbit and the periodic time may- 

 be easily effected; for we have only to find the equations ofc, y 

 and a from the equations 



cy^a 2 , 



c 2 _ fJU 

 7 2 ~~ // 



from which results 



c— 



c= 



a- ^ )l I 



a= -j to, 



p „ _ ^ 



-jl T> 7 ——K 1' 



fJu't—IJU* fl' 2 —/UL 2 



Also the velocity at either apse is given by the formula v= , _ NO , 



(a + cy' 



which gives = — — for the two limiting velocities. 



Again, the general expression for the time is 



. (« 2 + c 2 ) f 2ac . \ 



fJLt L « 2 + C 2 J 



Suppose, then, a planet to be describing an ellipse under the 

 attraction of the sun, and a fictitious body moving in a circle 

 described about its axis major to leave an apse simultaneously 

 with the planet, and that its velocity parallel to the axis major 

 always remains equal to that of the planet in the same direction. 

 Then the arc swept out by such body subtends at the centre the 

 angle which measures the eccentric anomaly of the planet, and 

 may be termed its eccentric follower. The motion of this ec- 

 centric follower may be physically produced by supposing it to 

 be attracted to two centres of force of proper absolute magni- 

 tudes and duly placed in the major axis, attracting according 

 to the inverse fifth power of the distance; this is an immediate 

 consequence from the preceding expression for /. 



