114 PARTICULAR CASES OF EQUILIBRIUM. 



We shall proceed to examine this particular case in some detail, 

 less for its own importance than as a useful transition to more com- 

 plicated problems. 



132. CYLINDRICAL SYSTEMS. Consider a uniformly electrified 

 unlimited line of density A, that is to say where the charge is X for 

 unit length. At each point of the dielectric, the force passes through 

 the axis, and is perpendicular to it. 



The flow of force proceeding from unit length is equal to 4?rA ; 

 at a distance r^ this flow traverses the lateral surface zirr of the cor- 

 responding equipotential cylinder, and the force F is defined by the 

 condition 



where 



(i) F-. 



The force is therefore inversely as the distance, as we have already 

 seen (80) for cylindrical condensers. The equation 



^ 



gives for the equipotential surfaces 



(2) V= -2\t.r + const, 



that is to say, a series of concentric cylindrical surfaces. Draw two 

 planes perpendicular to the axis and at the distance e ; the mass 

 which they include is m = Xe. The flow of force which will pass 

 between these two planes will therefore be 477^, and it is evident that 

 if we draw through the axis 47rw, planes making equal angles with 

 each other, each of the qicm dihedra thus determined constitutes an 

 orthogonal tube in which the flow of force is equal to unity. 



Take a plane perpendicular to the axis as the plane of the figure. 

 Let A (Fig. 28) be the trace of the electrified line, and Ax any axis 

 from which we shall count i, 2, 3. . . , the traces of the %xm planes 

 drawn through the axis. Lastly, let be the angle which the straight 

 line, number N, makes with the axis A#, it is manifest that the flow 

 of force corresponding to the angle is 



9 



Q = 47TW = 



