162 Dr. Hirst on Equally Attracting Bodies. 



number of surfaces, however, capable of attracting the pole with 

 the same intensity along the same line, we shall consider, solely, 

 the group whose corresponding elements attract equally. The ex- 

 pression ' equally attracting sui-faces ' being interpreted in this 

 very restricted manner, it will be seen that the problem before 

 us reduces itself to a purely geometrical one, which, apart from 

 the theory of attraction, is not without interest. 



3. Let dxT be the measure of the solid angle of a very small 

 cone having its vertex at the pole, in other words, let dcr be the 

 element which such a cone intercepts upon a sphere, with radius 

 unity, described about the pole as centre. If ds be the element 

 of an attracting surface intercepted by the same cone, r the di- 

 stance of the element from the pole, and ■<^ the acute angle be- 

 tween the radius vector and the normal to this element, then 

 ds . cos ■\lrz=r'^da, 



so that the attraction upon the pole, due to the matter spread 

 over the element ds with the density g, will be proportional to 



§ds gd<T ,,, 



r' cos ^fr' 



that is to say, the attraction of an element at any point of a 

 given surface is dependent solely upon the ratio between the 

 density at that point, and the cosine of the acute angle between 

 the normal and radius vector. 



From this we easily conclude, that when the densities at corre- 

 sponding points are equal, the pole is attracted with the same in- 

 tensity along the same line by all surfaces whose corresponding 

 elements {or tangent planes) are equally inclined to their common 

 radiiis vector, or, in other words, by all surfaces which cut at the 

 same angle each ray of a pencil whose centre is the attracted point. 



Of course the resultant attractions of two such equally attract- 

 ing surfaces will have like or opposite directions according as 

 their con'esponding points are situated on the same or on oppo- 

 site sides of the attracted point. Our solutions will always in- 

 clude both cases ; but since no difficidty will be encountered in 

 distinguishing between the two, we shall, for simplicity, keep 

 the former — where the surfaces attract in precisely the same 

 manner — mox'e especially in view. 



The equation (1) also shows that the pole is attracted with the 

 same intensity along the same line by all surfaces whose densities 

 at corresponding points are proportional to the cosines of the acute 

 angles between the common radius vector and the normals at those 

 points. 



The ratio between the density and the cosine of the acute 

 angle between the radius vector and normal, which according to 



