280 Trans. Acad. Sci. of St. Louis. 



For a = equation (2) becomes 



\co=7rax (14). 



This result is also obtained by making x = in equation 

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

 This line is the right section of the cylindrical surface, which 

 separates the lines of force terminating in the electrified line 

 from those which never reach it. In Fig. 3 the dashed line 

 whose vertex is / is the critical line. The critical line must cut 

 OY in a point at which the force due to the electrified plane 

 balances that due to the electrified line. For this point 



OI=r =}L (15). 



Inspection shows that (2) is the equation of a line of force 

 which is inside the critical line and (3) is the equation of a 

 line of force which is outside the critical line. 



For a) = 7r equation (2) becomes 



*,= ---- (16), 



a it a 



in which x 1 is the distance from O to the asymptote of a line 

 of force which is inside the critical line. For co = ir equation 

 (3) becomes 



®ii = - +*o (I 7 )' 



a 



in which x n — x {) is the distance between the two parallel 

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

 For a = equation (16) becomes 



sV=-=**o (18). 



o 



which is also obtained by making x = in equation (17). 

 Hence the distance from to the asymptote of the critical 

 line is x' 



