74 Prof. Sylvester's Astronomical Prolusions. 



circle ; call the distances of the centres of the images from that 

 of the image circle c, c' respectively. The points of contact of the 

 images with a common exterior tangent will be corresponding 

 points, and this tangent will pass through the centre of the 

 circle of reference; whence we easily derive (c 9 — r 9 ) {c' 9 —r 19 ) 

 = a\ and by similar triangles, 



Hence (c 2- r y=^« 4 . 



Whence, remembering that r must be less than c, we have 



c*-r 9 =-,a 9 , c 9 -r ,9 =-a 9 ; 

 c' c 



so that 



-t-S>M>-5) : 



Consequently, calling 1 { = +q, if F, G, two points in the 



diameter of the image circle, be distant c, c' respectively from its 



centre, and two cyclogenous forces ■ a ^_ L p tend 



(p 9 + qc 9 ) 6 ' (p l9 +qc 9 ) 3 

 to F and G, two such forces will serve to make a body describe 

 a circle, and, as we shall see, will be statically equivalent to a 

 single cyclogenous force tending to a fixed point, presently to be 

 determined f. 



It follows from what has been shown of any two correspond- 

 ing elements in the two figures, that the total vis viva contri- 

 buted by each at any moment of time, to the entire amount of 

 stand-up work in the revolving body is the same ; consequently, 

 confining our attention to one of the image circles, we see that 



v 9 oc r~ a ^2" Hence using u to denote, as before, the angle 



at the centre, we have 



du 



di 



which is of the form which gives the motion of a planet in 

 eccentric anomaly; consequently, by a proper adjustment of the 

 constants, the motion due to the cyclogenous centres F, G may 



* Calling F, G the two centres, F', G' the images of F, G respectively, 

 O the centre of the image circle, it is easily seen hat r 2 =FO . FG' r' 2 = 

 GO.GF. 



f The proof of this through the medium of the two circular images 

 requires —q to be employed, but the laws of analytical continuity allow 

 q to be made to change its sign. 



-t. cc a x + c 9 f qc 9 — 2ac cos u t 



