Prof. Sylvester's Astronomical Prolusions. 73 



going off in a straight line towards infinity on reaching such 

 point; but inasmuch as in either case an infinitesimal devia- 

 tion from the tangential direction will cause the remainder of 

 the orbit to be described, we may consider, in an analytical sense, 

 that the revolving body under the influences of such force de- 

 scribes the entire orbit. We may give the name of cyclogenous 



force to any central force of the form pq^ 3 , and, if we care 



to draw the distinction, call it internally cyclogenous or endo- 

 cyclogenous when the k is positive, and externally cyclogenous or 

 exocyclogenous when k is negative. If we call the cyclogenous- 



force-function V, so that — is the cyclogenous force itself, we 

 have, by integration, V=j . r-%—- 



(/° 2 ±*) 2 * 



Let us now proceed to calculate the attraction of a circular 

 plate (of radius r) of uniform density, whose particles attract 

 according to the law of the inverse fifth power of the distance 

 upon an external particle at the distance p from the centre. If 



we call this g~ , we have 



4 Jo J (r 2 + p 2 -2rp cos Of 



By comparison of * _ with the integral 



Jo (r + p — 2rp cos 6)~ 



which represents twice the area of an ellipse of excentricity, 

 " a -if a * we ^ n( ^ instantaneously 



p _7r^ 2(/3 2 -fr^r 7T gr 2 

 4 Jo (/o 2 -r 2 ) 3 ~4 (p*-r*f 



Thus P is of the form of the cyclogenous-force-function, so 

 that the force of attraction to the centre of a circular plate attract- 

 ing according to the inverse fifth power of the distance, upon an ex- 

 terior point, is an external cyclogenous force. From this we may 

 easily draw the conclusion that any circular orbit cutting ortho- 

 gonally a circular plate whose particles attract according to the 

 inverse fifth power of the distance may be described (or, at all 

 events, the concave part of it be described) by virtue of such force 

 of attraction. 



Let us now consider the joint effect of two such circular 

 plates, images of one another, lying one entirely within, the 

 other entirely without a given circle. The centres of two 

 such circles, it will be borne in mind, are not images of one an- 

 other. Let r, r' be the radii of the two images, a of the image 



