Dr. Hirst on Equally Attracting Bodies. 163 



this theorem must be the same for all surfaces and a determinate 

 direction of the radius vector, will, in general, vary with this 

 direction; in other words, it will be a function of the two 

 angles 6, ^ which determine the same. The theorem leads im- 

 mediately to the solution of the following problem : — To deter- 

 mine the density on a given surface in order that any portion of 

 the same may attract the pole in the same manner as the corre- 

 sponding portion of any other surface with respect to which the 

 density is known. In the present memoir, however, instead of 

 entering into the details of this problem, we shall occupy our- 

 selves exclusively with the consequences of the first theorem. 



4. A few of the simplest of these consequences may be at once 

 deduced. For example, each ray of a pencil is cut twice, at the 

 same angle, by a sphere ; so that the centre of the pencil, if it 

 be without the sphere, will be attracted in the same manner by 

 corresponding portions of the convex and concave surfaces; and 

 if the centre of the pencil be within the sphere, it will be kept in 

 equilibrium by the attractions of corresponding portions, pro- 

 vided that, in both cases, the densities are the same at corre- 

 sponding points. The well-known theorem given by Newton in 

 the 70th Prop, of the first book of his ' Principia,' is a pai-ticular 

 case of the above. 



5. Surfaces whose like-directed radii vectores are proportional 

 to each other, or, as we shall call them, similar surfaces, also at- 

 tract the pole in the same manner when the densities at corre- 

 sponding points are equal. This is at once evident from the 

 equation (1), which shows that the attraction of an element inter- 

 cepted by a given cone is independent of its distance from the 

 pole. If r = cf{e,(f)) 



be the equation of any surface, those of all surfaces similar to it 

 will be found by giving to the constant c all possible values. 



6. It can easily be shown, too, that, under the same hypo- 

 thesis with respect to the densities at corresponding points, 

 inverse surfaces, or those whose like-directed radii vectores are 

 inversely proportional to each other, also attract the pole equally. 

 Although the truth of this assertion will be at once manifest 

 from the general equation of equally attracting surfaces, to be 

 hereafter given, it may be well to give at once a simple and 

 direct demonstration of tlie same. Let m and ?«, be two conse- 



cutive points on a given surface, m' and ?«', their corresponding 

 points on the inverse surface, and the attracted point or pole; 



M 2 



