430 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Problem IV. 



To fin.l the distribution at the edge of a plate of finite width, which is 

 at potential v, near to and parallel to an infinite plate at potential zero. 



i = -oo 



oo t = a 



CO 



A 



V = V 



V B 



t = 



C t = 1 



CO 00 



I) + CO 



v = 



The Schwarzian transformation then becomes 



C t — c 



When a = b this assumes the form 



V(t - a) (< -_c) dL (a) 



rf* = ( 1 + ■ rf<. 



< — c V * 



-:) 



c 



— [t + (c — o) log (if — c) + const.]. (b) 



To integrate the above expression (a) make the substitution t, — c — v, 



then t — a = v + c — a = v -\- d, d = c — a, 



t — b = v + c — b = v + e, e — c — b, 



and d -\- e = 2c — (a + b), de = (c — a) (c — b) 



(a) now becomes 



rfz V(v + rf) (v + e) 



— = — * ^ '- dv 



C v 



and is of the form under the radical of 



X = a + f3 x + y x 2 = rfe -f- (d + e) y -+- r 2 , 

 where a = rfe, /3 = d + e, y = 1. 



Using Byerly's tables of integrals we find that 



