384 VIII. NEWTONIAN POTENTIAL FUNCTION. 



Apply the above transformation to cylindrical coordinates 



g = 1, 



a / aF 



aF 



a /i aF 



46) 



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

 homogeneous body with generators parallel to the axis of s and of 

 infinite length, the potential is independent of 8 and satisfies at 

 external points, ^ v 



= Vi- 



cx z 



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

 the ordinary differential equation 



dg 



= or 



48) 



The force in the direction of Q 

 is inversely proportional to the first 

 power of Q. 



We may verify this by direct 

 calculation. Let us consider the 

 cylinder as infinitely thin, with cross- 

 section cb. We will find the com- 

 ponent of force in the direction of p. 



The action of dm at 8 on P at 

 distance Q (Fig. 137) is 

 dm dm 



Fig. 137. 



The component parallel to Q is 



dm 



COS 5 = 



Now since, calling the density d, 

 total force in direction 



49) 



d&d0, we have for the 



