﻿470 Mr. W. M. Hicks on the Self-induction 



9. The same method of attack can be employed in dealing 

 with gravitation potentials. Here 



-V 2 </> = -47TO-, 



where a is the density at the point (x, y, z). 



If / be a particular solution of this, and <j> denote the 

 general solution of V 2 </> = 0, then at all points within the 

 attracting body the potential = /+ fa, and at all points out- 

 side = fa. Also fa and fa must be finite and continuous in 

 their respective regions, and at the surface of the body the 

 fa and fa and their first differential coefficients are continuous. 

 These conditions suffice to determine fa and (j> 2 . 



Take, for instance, the case of a uniform sphere. Inside 

 a is constant, and 





d*V 2 dV 

 dr 2 r dr 



= — 



47TO\ 



A particular integral is 



3 



7rcrr 2 



5 



whilst the 



i general solution is 









V = A + 



B 

 r 





Hence: 



Inside, </> = — 



2 

 3™ 



• 2 + A- 





Outside, </>' = A' 



r 





Inside, 



(j> is finite, .*. B= 



= 0. 





At the 



surface, when r=a, 

 (f> = <f>', and 



dcj> 

 dr '' 



dfa 

 ~ dr' 



Therefore . 2 2 .. B^ 



6 a \ 



4 b' r 



-3™ a = ~? j 

 whence A' = A — 27rcra 2 , 



4 

 3 



and 



<f/ = A — 27r<ra 2 + o . 



o r 



