588 PROF. J. J. THOMSON- AHD MR. G. F. C. SEARLE OH THE RATIO OF THE 
Correction for the want of equality in the distrihution produced hy the airspaces 
hetween the inner cylinder and the guard rings .—To find this correction we shall find 
the distribution of electricity on the system represented in the figure, when AB is 
the section by the plane of the paper of an infinite horizontal metal plane, and 
CDE, FGH sections of conductors, CD and GH being’ horizontal and at the same 
distance from AD ; and DE and EG vertical. Let h be the distance between the 
planes CD, and AB, and 2c the breadth of the slit DE EG. 
Let us take AB as the axis of x and the vertical line midway between ED and EG 
as the axis of y. Then writing 2 : for x + vy, and supposing that (f) and xjj are the 
stream and potential functions respectively, we find by using Schwarz’s method 
that the solution of the problem is given by the equations 
dz = — A 
{1 
f - ah 
dt. 
rt < 1 
( 1 ) 
(f) Lxjj = log 
t — a 
t -)“ Q- 
t being supposed to hare all real values from — qo to + 00 . 
Eor putting t — sin 6, a = sin a, and integrating (l) we find 
Z =■ K [6 — h cot a log 
\ 2 sin (a + 0 ) 
or 
/ , sin (a — 
X iy = Kid — h cot a log ——- 
( 2 ) 
(3) 
as 0 goes from 0 to a. the right hand side of this equation is real so that y = 0 and x 
ranges from 0 to + GO, this gives the positive half of the plane AB. As 9 goes from 
a to ^77 
sin {9 — 
sin (« + 
X ly = Kid — -g cot a log 
d 
d) 
+ h 177 cot a 
