Prof. Sylvester's Astronomical Prolusions* 71 



sole necessary and sufficient condition for any determinate orbit 

 being described, that, if several sets of forces taken separately 

 can make a body describe a certain path, then all the sets act- 

 ing collectively will make it describe the same path, provided 

 the vis viva at the starting-point, on the latter supposition, is 

 the sum of the vires viva on the former one. 



Suppose, now, a zone of matter bounded by two arbitrary 

 contours P, Q to lie anywhere within a circle C, and another 

 zone bounded by two contours P', Q', the geometrical inverses 

 (or reciprocals) of P, Q to lie outside the same. Then these 

 two zones may be divided into corresponding rectangular ele- 

 ments by transversals drawn through the centre of the circle, 

 points being taken all along every radius of one zone, and cor- 

 responding points along the radii of the other. If r, r 1 be the 

 distances of the centres of any two corresponding elements thus 

 obtained from the centre of the circle, d6 the angle between 

 the two transversals which pass through both pairs of points in 

 both figures E, E', the areas of the respective elements will be 



dd.dr.r;d6{-dr')r',i.e. -d0d(j\ji 



so that 



E _r 4 _r 2 



Hence if the densities of E, E' be the same, and they attract with 

 forces varying as the inverse fifth power of the distance, they will 

 serve to make a body describe the circle in question, E, E' taking 

 the place of /x, /jf and r, r of c, 7 in our previous formulae ; and 

 as this is true of each pair of elements, it will be true of the 

 two entire zones which they compose, the law of density being 

 perfectly arbitrary, except that it must be the same for corre- 

 sponding points in the interior and exterior zone. The contour 

 Q may be made to coincide with Q' at the circumference of C 

 if we please ; and then, as a particular case of the proposition 

 above, we may suppose the united zones to consist of homoge- 



two central forces). Call P, P' the squared perpendiculars on the tangent 

 from the two centres respectively, (j) an arbitrary function of any affection 

 of the position of the revolving body (ex. gr. of the length of arc or radius 

 of curvature at any point), then 



P ' p' 



will be the general system in question. When the stored-up work for 

 each point in the orbit is known, the radial equation gives the central 

 forces without integration. Thus, ex. gr., if a body move in an ellipse with 

 uniform velocity acted on by forces towards the foci, the equation in 

 question shows that they are equal, and vary as the inverse square of the 

 conjugate diameter. 



