72 Prof. Sylvester's Astronomical Prolusions. 



neous matter, or, if we please, of matter whose density at any 

 point is only a function of the angular position of the line 

 joining the point to the centre of the circle. Thus, if we suppose 

 a plate of matter of uniform density and of indefinite extent, and 

 attracting according to the law of the inverse fifth power, a point 

 anywhere placed upon it may be made to move in any desired 

 circle under the influence of the plate's attraction, if we cut away a 

 portion of the plate surrounding the centre of such circle, and 

 leave a proper margin exterior to the circle — the rule being that 

 the intrados of the figure so obtained maybe of anyform whatever, 

 provided the extrados be its electrical image or inverse. The 

 initial velocity to be communicated to the moving point will of 

 course be determined by the form of either of these bounding 

 curves. 



It is hardly necessary to add that instead of a zone we may 

 take a patch of matter bounded by a contour of any form 

 within. the circle C, and then, finding the inverse of this contour 

 so as to obtain a corresponding external patch, the two together, 

 by the combined attractions of their particles according to the 

 inverse fifth power of the distance, will serve to make a body 

 describe the circle C ; and conversely, since any two circles may 

 be made reciprocals (inverses) to each other by duly determining 

 the centre and radius of the circle of reference, it follows that 

 any two circles of matter attracting according to the above law, 

 will serve to keep a body moving in a certain third circle. 



By calculating the attractions of these two circular images, 

 and replacing them by forces tending to their centres, we shall 

 be able to transform and generalize the results previously ob- 

 tained. But first it will be expedient to recall attention to the form 

 of the single central force which serves to make a body describe 

 a circle. We have found that such force, when the centre lies 



within the orbit, is of the form 2 z.\3 > an( ^ ^ * s eas y *° see 



that when external thereto, it takes the form , Q 7to 3 — in either 



case k being the product of the two distances of the force-centre 

 from the extremities of the diameter drawn through it ; when 

 the force is external, this product is the square of the tangent 

 drawn to the circle from the centre. At the points of contact 

 the force and velocity both become infinite, and the latter 

 changes its sign. 



In a physical sense, only the concave part of the circle will 

 be described by virtue of attraction to the centre, the revolving 

 body going off in a straight line towards the centre when any 

 point of contact is reached, and in like manner only the convex 

 part by virtue of the repulsive force from the centre, the body 



