324 ELECTROSTATICS. [PT. II. CH. VII. 



In fact we see from the equation that for any finite u' , v must be 

 zero when x is zero. 



The curves y = const, are different in appearance, on account of 

 the minus sign, u has minima for v'=Q, TT, 2-Tr... having the 

 values u = log cos y which are all negative, and decrease more and 

 more rapidly as y increases to Tr/2. The maxima of u' are how- 

 ever infinite. In fact while u increases continuously as v varies 

 from to 7T/2, as soon as v > y the parenthesis becomes negative 

 and u is imaginary. The curves y const, accordingly approach 

 horizontal asymptotes v' = y. These curves correspond to the 

 hyperbolas of the last figure, the sinuous curves corresponding to 

 the ellipses. Corresponding regions in the three figures are simi- 

 larly shaded. The circle in Fig. 66 corresponds to the vertical V- 

 axis in Fig. 67. 



If we choose for the functions V and ^ the values Vy, 

 ty = x, and consider the strip between v' Tr/2 and v = w/2, we 

 have the case of the electrification of an infinite plane with a free 

 edge, lying between two infinite planes parallel with it at distances 

 7T/2 from it, and extending to infinity on all sides. Since at a 

 distance from the edge x is equal to u + log 2, the field is straight, 

 but the charge from the edge to the point u' is greater by 

 (log 2)/47r than if the plate extended to infinity instead of stop- 

 ping at the edge. Thus the edge increases the capacity of the 

 upper or lower side of a portion of the plate of any width by the 

 amount K = (log 2)/47r ( F 2 - F a ) = (log 2)/27r 2 . This result may 

 be used to find the capacity of a circular plate between two infinite 

 parallel plates at a distance from it d so small that the edge of the 

 circular disc may be considered straight *. The effect of the edge 

 is the same as that of increasing the radius by (log 2) d/(ir/2), so 

 that the capacity would be, counting both sides, 



d ( lo 2 V 7r _ & 2 



* Maxwell, Treatise, Vol. i. Art. 196. 



