Boever — Geometrical Properties of Lines of Force. 281 



74 



For (o= a equation (3) becomes x — x^ 



- ~ ; from equa- 



x„ = -. 



This shows that a line of force out- 



tion (17) ajy 



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 O, as before, 

 be the trace of the electrified 

 line and AB the trace of the 

 electrified plane. Through O 

 draw OX parallel to AB and 

 OY perpendicular to AB. 

 At any point P draw the 

 arrows PE — 27r<r and PF 



2X . u ^ 



= — 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, OD z=z x 

 and ^ TOP = o the slope of PQ is 



FlO. 4. 



ax 



HP 



PE — FH 

 HP 



2X 

 27r<T cos o) 



r 



2X, . 

 — sm 0) 



r 



^ , y y 



But cos <y = - 



r- l/x2 + 



y 



and sin a> = - = 



»• l/jc^ + 2/2 ' 



therefore 



„_dy _ Xy — -rra (a;^ + y^) 

 dx Xa; 



(19). 



When integrated this expression becomes 



X arc sin 



X 



Vx^ + y2 



— airx = C 



