58 



Electrostatics Field of Force 



[CH. n 



where 6 l is the angle QP^. Thus the surface integral of normal force 

 arising from e 1} taken over the circle QN, is 



27T6! (1 - COS 00 



and the total surface integral of normal force taken over this surface is 



1 -cos - 



If we draw the similar circle through Q\ we obtain a closed surface 

 bounded by these two circles and by the surface formed by the revolution 



Q' 



FIG. 23. 



of QQ'. This contains no electric charge, so that the surface integral of 

 normal force taken over it must be nil. Hence the integral of force over 

 the circle QN must be the same as that over the similar circle drawn 

 through Q'. This gives the equations of the lines of force in the form 



(integral of normal force through circle such as QN) = constant, 

 which as we have seen, becomes 



S0! cos O l constant. 



Analytically, let the point Pj have coordinates a 1} 0, 0, let P 2 have 

 coordinates a 2 , 0, 0, etc. and let Q be the point x, y, z. Then 



cos 6 l = 



X X, 



and the equation of the surfaces formed by the revolution of the lines of 

 force is 



v e l (x- tfj) 



2 -= = constant. 



It will easily be verified by differentiation that this is an integral of the 

 differential equation 



Sy_Y 



cix~ X' 



