Moever — Geometrical Properties of Lines of Force. 293 



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

 critical line. 



For CO =2- equation (27) becomes irx^ — ttx^^ = — ; from 



2m 

 equation (41) irx^^ — ttx^^ = — . 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 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 



the arrows PU = ^ira and PF = —f representing the forces 



due to the electrified plane and the electrified point respec- 

 tively. On PU and PP 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 P§ is 



, „^ ^ira -„cos (o 



a _dy _ QH _ PE — FH _ r^ 



But 



dx HP HP m . 



-TT sm ft) 



y y , X X 



cos ft) = — = , „ r and sm w = — = 



Therefore, 



3. 



dx mx 



When integrated this expression becomes 



(-1 + 7^0 + ^"^' = ^ 



m . - . 



Vx^ +y' 



or 



m versin co — irax"^ = O, 



