Boever — Geometrical Properties of Lines of Force. 293 



H<'nce x\ is the distance from O to the asymptote of the 

 critical line. 



For G> ^ g- equation (27) becomes irx* — irx^ = — ; from 



2m 

 equation (41) ttx/i — 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 O represent the electrified point and AB the 

 trace of the electrified plane. Through draw OX parallel 

 to AB and OT perpendicular to AB. At any point P draw 



m 

 the arrows P£J = iira and PF = — j- representing the forces 



due to the electrified plane and the electrified point respec- 

 tively. On PE and Pi^ 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 



8 = 

 But 



27r<T _ cos a» 



_dy _ QH _ PE — FH__ r^ 



dx HP HP m . 



^8m« 



y y XX 



cos o) = — = . ^ and sin « = — = . ■ 



Therefore, 



o _ ^ _ my — '^(T (x^+y^y .^^^ 



dx mx 



When integrated this expression becomes 



y 



Vx'^ + y 



H-' + Tm}^'""'^'' 



or 



m versin (o — irtrx^ = C, 



