Roever — Geometrical Properties of Lines of Force. 281 



77 



1 X 



For a) = g equation (3) becomes x — x = r; - ; from equa- 

 tion (17) x n — x = -. This shows that a line of force out- 

 side the critical line and the two asymptotes of this line of 

 force cut two equal intercepts from the line OX, which 

 passes through and is parallel to AB. 



In Fig. 4. let 0, as before, 





Q 



VH 



P 



U 



~T~ 



be the trace of the electrified 



line and AB the trace of the 



electrified plane. Through 



draw OX parallel to AB and 



— X OY perpendicular to AB. 



At any point P draw the 



arrows PE = 2ira and PF 



2X 

 = — representing the forces 



due to the electrified plane and the electrified line respec- 

 tively. On PE and PF construct the parallelogram PEQF; 

 the diagonal PQ will represent the magnitude and direction 

 of the resulting force at P. PQ is the tangent to a line of 

 force at the point P. Then if OP = r, PD = y, OB = x 

 and ^L TOP = co the slope of PQ is 



Fig. 4. 



Q _dy_ QH = 

 * " dx HP 



PE — FH 

 HP 



2X 



277(7 COS (O 



r 



2X 



sin a) 



y y 



But cos to = - = 



r Vx 2 + y 2 



x x 



and sin to = - = 



»' Vx 2 + y 2 ' 



therefore 



8 = 



dy Xy — 7t<t (x 2 + y 2 ) 



dx Xx 



When integrated this expression becomes 



x 

 X arc sin ■ , „ = — ottx = C 

 vx 2 4- y 



(19 



