On Potentials ara\ tlieir AppUcation. 163 



importance of this Function by a simple explanation of 

 its nature and use. 



As a further example, we will apply it to finding the 

 attractions of a Sphere, on an external or internal point. 

 This, it is true, is a case in which it is easier to integrate 

 directly the expressions for the attractions ; but we use it 

 merely as an example. 



"We already know by other methods that these attrac- 

 tions will be 



r' 

 for an external point = | Trp . — 



" " internal " = 2 rrp r^ + 1 7rp . x 

 where r = radius of the sphere. 



X = distance of attracted point from centre, 

 p == density of the sphere. 

 Let us get these values through the Potential Function 

 in each case. 



If r, j[<, CO be the polar co-ordinates of any surface ele- 

 ment of the Sphere, referred to the centre of the Sphere 

 as origin ; — r being the radius ; jtt, the cosine of the latitude 

 (or better, the co-latitude) ; w, the longitude of the element ; 

 p, the density of the sphere ; then the 



Mass of this element = p r* dr djw dw. 

 If in Fig. 2, x' = distance of this element from p, then 

 will, by ordinary Geometry, 



