364 



YE!. NEWTONIAN POTENTIAL FUNCTION. 



Now 



Differentiating , keeping r 

 constant, 



sds = 



and introducing s as variable 

 instead of -9-, 



Fig. 128. 



If P is external we must integrate first with respect to s from 

 h r to h r. 



77) 



A r 



Hence the attraction of a sphere upon an external point is the 

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



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 JR t and outer R 2 , the limits for r being JR 1? R 2 , 



B, 



78) 



M 



h' 



We have 



dV 



M 



dh* 



h 8 



If, on the other hand, P is in the spherical cavity, h < JR 17 the 

 limits for s are r h, r -\- h 



R 2 r + h R. 2 



V= ^ CCrdrds = 4=*$ Crdr 



79) 



which is independent of fc, that is, is constant in the whole cavity. 



/} V 



Hence = 0, and we get the theorem due to Newton that a homo- 



