llieory of the Quadrant Electrometer, 245 



The solution of the ?6'-problein can be effected by the 

 transformation 



B(/> + 4>/r iB 



B^y ~ \/{io^ — l){w-h){w-\-cy 



where and i|r are the potential and stream- function re- 

 spectively. 



The latter transformation is reducible by means of elhptic 

 functions. When we write the condition that the finite 

 plate is at unit potential, while the semi-infinite plates are 

 at zero potential, we readily find that the charge on the 



4K' 

 finite plate i^— tt^ , where K^ and K are the usual quarter 



periods of the elliptic functions, and the modulus is given by 



and (,V + \^2) (6 - c) - 2 \V(1 - he) = 0. 



Let ns put 5 = 6o + So 



and take the root 



X, Wo ~ 2So ^' 



we get k-=i-^{^^y. 



If Sq be small we get 



Let ho be the root of 



2 *=i-V""a'-l~2-8 • 

 Then we find from equations (10) and (11) 



and 



On -kCS / O . 7 /9 i»\ 1 



and hence 



,, ,.,,,_ l7rV(l-V^)( a'^-l)^ 

 2 82 " (a^ + V'-2)3 



