156 THEORY OF NEWTONIAN FORCES. [PT. I. OH. IV. 



dV M 



Wehave ~dh=-> 



_ 



dh* ~ h s ' 



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

 the limits for s are r h, r+h 



_ 



V = 



. , 



rdrds 



f ^ 



= 4?rp I 



JE, 



rdr 



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

 Hence ^j- Q> an d we get the theorem that a homogeneous 



spherical shell exercises no force on a body within. (On account 

 of symmetry the force can be only radial.) 



If P is in the substance of the shell, we divide the shell into 

 two by a spherical surface passing through P, find the potential 

 due to the part within P, and add it to that without, getting 



(Bf - 



dV_4nrp{Rf } 



dh " 3 U 2 J ' 



d*V_ iirppRf . ,| 

 dh,* ~ ' 3 1 A j ' 



Tabulating these results 



dV 

 dh 



d^V 



dh* 



2,7rp (R* - 

 

 



c "r / T> 3 7? 3\ 



OL V-"^ ~ -"1 / 



3_ 



