292 Trans. Acad. Sci. of St. Louis. 



For a = equation (26) becomes 



m versin to = Trtrx^ (38). 



This result is also obtained by making Xq = in equation 

 (27). This is the equation of the limiting or critical line. 

 This line is the meridian curve of the surface of revolution 

 which separates the lines of force terminating in the elec- 

 triOed point from those which never reach it. The dashed 

 line in Fig. 6 is the critical line. For the point / in which 



the critical line cuts OY. — = ^tto or 



' -..2 



«^='-» = V2^ (^«>- 



'27r<T 



Inspection shows that (26) is the equation of a line of 

 force which is inside the critical line and (27) is the equation 

 of a line of force which is outside the critical line. 



For ft) = TT equation ( 26) becomes 



m (2 — versin a) 

 x,^=—^ ^ (40), 



in which x^ is the distance from O to the asymptote of a line 

 of force which is inside the critical line. For ft) = tt equa- 

 tion ( 27) becomes 



2w 



in which (x^, — x^) is the distance between the two parallel 

 asymptotes to a line of force which is outside the critical 

 line. 



For a = equation (40) becomes 



12m 

 ^^ = V^ = 2^0 (42), 



which is also obtained by making x^ — in equation (41), 



