430 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Problem IV. 



To find the distribution at the edge of a phite of finite width, which is 

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



t = — CO For generality put at 



A ti z= a, 



t-0 





C ^ = 1 I, . 7 3 



TT 



B t.^ = b, ag — , 



CO ^ ^00 ^ 



I) + 00 



v = C ts — c, as =z 0. 



The Schwarzian transformation then becomes: 



O t-c 



When a — b this assumes the form 

 dz t — a 



dz _ V{t - a) (t -c) ^^^ ^^^ 



c t — c 



( c — a\ 



dt—[ 1 + dt 



c 



\t + (t- — a) log {t — 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 — i = i" + c — b = V + e, e = c — b, 



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



(a) now becomes 



dz \/{u + d) (v + e) , 



— = -^ '-^ dv 



6 V 



* 



and is of the form under the radical of 



X = a + (3 X -{- y x^ ^ de + (d + e) v + ^•^ 

 where a = de, fS := d + e, 7 = 1. 



Using Byerly's tables of integrals we find that 



