174 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. 



(7) _ . 



dr 2 r dr dr \ dr J 



_ _^ 



dr '' } dr ~r 2 ' 



V=-- + c'. 

 r 



But since Lim (rV) = M, 



r=oo 



Lim [ c + c'r] = M, 



T= 00 



we must have c' = 0, c = M. 



Apply the above transformation to cylindrical coordinates 

 /L = 1, h = 1, h.,. = - , 



p 



_ idv 



" 8? + dp 2 + p dp + p" 2 9 2 ' 



If we apply this to calculate the potential due to a cylindrical 

 homogeneous body with generators parallel to the axis of z and 

 of infinite length, the potential is independent of z and satisfies at 

 external points, 



- 



__ 18F 1 8^F 

 dp 2 p dp p 2 do)- ' 



If the cylinder is circular, V is independent of &>, and we have 

 the ordinary differential equation 



1 dV 

 +- j =0, 



a/5 2 p dp 



dp p' 



d (. dV\ 1 

 ' l g-^T =-- 



log -7- = - log p + const. 



dp 



