Roever — Geometrical Properties of Lines of Force. 293 



Hence x\ is the distance from O to the asymptote of the 

 critical line. 



t m 



For &) =«r equation (27) becomes 77-r 2 — 7rx 2 = — ; from 



2m 

 equation (41) ttx^ — itx 2 = — . This shows that a line of 



force outside the critical line and the two asymptotes to this 

 line of force are meridian curves of co-axial surfaces of 

 revolution which cut from a plane OX (through O and 

 parallel to plane AB) two annuli of equal areas. 



In Fig. 4. let represent the electrified point 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 



m 

 the arrows PE = 27ro-and PF = — representing the forces 



due to the electrified plane and the electrified point respec- 

 tively. On PE and PF construct the parallelogram PEQF; 

 the diagonal PQ will represent the magnitude and direction 

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

 Then from the figure the slope of PQ is 



o = fy _ _ QH __ PE — FH 



m 



— cos &) 



But 



dx HP HP m 



—^ sin a) 



y y x 



cos &) = — = .. _ r and sin &> = — = 



r Vx 2 + y 2 r Vx 2 -\-y i 



Therefore, 



a. 

 dy = my — ira (x 2 +y 2 ) 2 ,^ 



dx rax 



When integrated this expression becomes 



m 



{- l+ vm) +wa * =0 



V x 2 + y 

 or 



ra versin &> — irax 2 = (7, 



