Roever — Geometrical Properties of Lines of Force. 281 



TT 



1\ 



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



\ 

 tion (17) ajjj — 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 O-X", which 

 passes through and is parallel to AB. 



In Fig. 4. let 0, as before, 

 be the trace of the electritied 

 line and AB the trace of the 

 electrified plane. Through 

 draw OX. parallel to AB and 

 OY perpendicular to AB. 

 At any point P draw the 

 arrows BE — lira and PF 



= — representing the forces 



X 



Fig. 4. 



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, PB z=y, OD = x 

 and ^ YOP = a) the slope of P^ is 



ax 



QH _ PE — FH 



2X 



lira cos 0) 



r 



HP 



HP 



2X 



sm oj 



But cos (0 = - 

 r 



y . . X 



-y===== and sin w = - 

 Vx^ + 2/2 r 



X 



Vx^ + y2 ' 



therefore 



^^dy ^ \y — TTa (x^ + i/) 

 dx Xic 



(19), 



When integrated this expression becomes 



X arc sin 



X 



Vx^ + 2/2 



aTTX = O 



