Prof. Sylvester's Astronomical Prolusions. 75 



be made identical with the motion in a circle of radius a with 

 centre at O, about the single cyclogenous-force centre at S. 

 Call ae the distance of S from O, M the absolute force at S, 

 then, comparing the vis viva on the two suppositions at the two 

 apsidal points, and again availing ourselves of the law of equal 

 production of vis viva from the two force-centres F, G, we obtain 



2Xc 2 _ M 



({a + c)* + qc*y '" ((« + «c) 2 +(« 2 -aV)) 2 

 (a — c) 2 -t-qc* a — ae 

 HenCC (a + c)* + qc*) = ~aTa7 



or (l-f-?)c 2 - — c+« 2 = 0. 



Calling c, c' the two roots of this equation, we have 

 1 j[_.J_ 



c + c' 2ae> 

 or, which is the same thing, the points 0, F, S, G form a 

 system of four points in harmonic relation. 



Hence, if we take a system of points, F, F', F" . . . G, G', G" 

 .... in involution, the double points of the system being at S 

 and 0, the cyclogenous force at S will be statically equivalent to 

 two cyclogenous forces directed to any two corresponding points 

 F, G. 



It is possible that this theorem may be modified so as to 

 admit of further generalization, and be made to extend to an 

 arbitrary system of points in involution, without regard to the 

 condition of being one of the double points; but I have not 

 had time to consider this point. 



In the particular case where F, G become images, in respect 

 to the circular orbit annexed to the force at S, the cyclogenous 

 centres F, G become centres of attraction, following the law of 

 the simple inverse fifth power, as already found. Since in all 

 cases the absolute forces at F, G are proportional to the squares 

 of their distances from 0, if we make cc' — a 1 " 2 , and draw the 

 circle whose centre is at and radius is a ! , and take two 

 figures, images of one another in respect to this circle, by the 

 same reasoning as applied to the case of a' = a it may be proved 

 that, provided the densities at corresponding points of such 

 images be the same, and the particles attract according to a certain 

 fixed cyclogenous law, their joint action will support a body in a 

 circular orbit whose radius is a and centre at 0. We might 

 again assume two such images to be circular, calculate the law 

 of attraction towards the centres according to the supposed law, 

 and so return to a new system of conjugate points replacing 

 F and G ; but I have not had time to ascertain whether such 



