79, 80] NEWTONIAN POTENTIAL FUNCTION. 155 



SO. Examples. Potential of a homogeneous Sphere. 



Let the radius of the sphere be JK, h the distance of P from its 

 center, 



r-JT/J*. . 



Let us put s instead of r, using the latter symbol for the polar 

 coordinate, 



Now s 2 = A 2 4- r 2 2hr cos 6. 



FIG. 34. 



Differentiating, keeping r constant, 



sds = hr sin 0d0, 

 and introducing s as variable instead of 0, 



We must integrate first with respect to s from h r to h + r, 

 if P is external ; 



h-r 

 M 

 3h ~ h' 



Hence the attraction of a sphere upon an external point is 

 the same as if the whole mass were concentrated at the oenter. 



A body having the property that the line of direction of its 

 resultant attraction on a point passes always through a fixed point 

 in the body is called centrobaric. 



If instead of a whole sphere we consider a spherical shell of 

 internal radius R^ and outer R 2 , the limits for r being R lf R 2 , 



