70 Prof. Sylvester's Astronomical Prolusions. 



It may be worth while pointing out a somewhat singular 

 consequence of the laws that have been above established for 

 the motion of a body in a circle about two reciprocal points 

 as centres of force. It is an immediate and now well known, 

 although for a time singularly overlooked, consequence* of the 

 linear form of the equation (2/cos£)p = C — 25) [dr(f) [where 

 / is any central force, and i the angle which it makes with p, the 

 radius of curvature at any point], which equation f exhibits the 



posing the equation above written into 



integrating the two equations, and making suitable substitutions, thence 

 results 



* See an article by M. Serret among the valuable notes of M. Bertrand's 

 edition of the Mecanique Analytique. The principle referred to must be 

 taken with analytical latitude, or the range of its application will be unduly 

 restricted. For instance, it is well known and easily demonstrable that a 

 body starting from rest in a position where it is equally drawn by two 

 forces converging to centres attracting according to the law of nature, 

 will oscillate in the arc of an hyperbola. Here the principle seems in- 

 applicable ; for the hyperbola will be concave to one focus of attraction 

 and convex to the other, but a curve actually described about either focus 

 would be concave towards it. But in fact the principle does apply ; for, 

 analytically speaking, any conic whatever may be described about an attrac- 

 tive centre of force varying as the inverse square ; only if it be convex to 

 the centre of attraction, its vis viva will be a negative quantity, and the 

 motion imaginary. In the case above supposed, the vis viva due to each 

 centre of force ending singly will be equal, but with contrary signs, so that 

 the body in such position must be supposed to be at rest ; then, by virtue 

 of the principle enunciated, it will for ever continue to move in the hyper- 

 bola, in which it would move really under the influence of one centre, 

 imaginarily under that of the other, — the imaginary motion blended with 

 the real continuous one changing the character of the latter into a re- 

 ciprocating movement, which is in no way contradictory to M. Serret's 

 theorem, which only determines the locus, but not the direction of the 

 movement at any point. 



f I am informed by the highest authority, the author of Reports on 

 Mechanics, which have become classic, that he has never seen this equa- 

 tion anywhere before employed. It is of course an obvious generalization 

 of Newton's rule, connecting the velocity with that due to a single central 

 force acting through one-fourth of the chord of curvature. As it springs 

 from a combination of the law of vis viva with that for centrifugal force, 

 I propose to call it the Equation of Radial Work. By aid of it, it is easy 

 to establish the following theorem, giving the most general binary system 

 of forces acting to hvo centres, which will make a body describe any given 



orbit. Call V, V the respective for cef unctions (so that -y-j -~rj are the 



