﻿Lines of Force and Equipotential Surfaces. 835 



Thus there are n sheets of the surface intersecting at P, but 

 these do not make equal angles with each other. The axis 

 of symmetry cannot be one of the equipotential lines, for 

 V n (fjb) cannot vanish for = or it. When n is odd yu. = 

 is a root of P n (/6) = 0, and therefore when an odd number 

 of sheets intersect one of them is normal to the axis of 

 symmetry. 



The force-function corresponding to (10) is well known 

 to be * 



A d 



-\\r= .r n+l {l—fj?) j- P n (/ji) + higher powers of r. 



The lines of force through P are in the directions given 

 by ^—±1, i- £• the two parts of the axis of symmetry, 



together with the roots of —V n (fju)— 0. This equation has 



one and only one root between any two consecutive roots of 



1%. 1. 



The Intersection of Two Linei 3 of Force and Two Sheets of an 

 Equipotential Surface. 



V n (fi)=0, and hence it has n-l roots between /6=±1 

 exclusive. The axis of symmetry is, therefore, always one 

 of the intersecting lines of force; there are w — 1 others 



* Lamb, 'Hydrodynamics,' p. 120. 

 3H2 



