22 
Transactions of the Royal 8ociety of South Africa. 
where iv — JJ -\- ?'V, U being the current function, and Y, the potential func- 
tion, gives the solution when the plate ?/ = Ctt is at potential o, and the 
plate y = o, is at potential D-tt. 
The elimination of between (1) and (2) gives — 
X = C(//" cos - 1 - ^d) . . . (3), 
r.nd y = C(//^ sin - ^ 'd + ^) • • • (4), 
v/hen Y = 0, y = C-^, 
and when Y = Dtt, y — o 
Now, by the principle of images, the distribution on the upper plate is 
the same as if the lower plate were a semi-infinite plate, y = — C-^r, and the 
differences of potential between them 2D7r. 
(3) Electrostatic Deflectio7i of the Cathode Ray. 
If a particle of mass on carrying a negative charge is projected from a 
})oint on the axis of x with velocity v parallel to x, the equation of motion is — 
^ - - eY- ^^^^ = e 
dt" dy dx ' 
, rtiv- d~f f/TJ 
or approxmiateiy, = , 
where Uq is the value of U, where ^ = o. 
mv" dy 
e dx 
dy 
dx 
but X = C(/S - log 1 -f f3), and U = D log 1 + B, 
therefore — 
y = CD f^-ljj logTT^ rf/i - U„y (h, 
7/ = CD 
1 + log 1 + /8 - 1 - /3 - I log^ 1 4- 
A. 
X 
- J/ 
0 
( 4) Cathode Ray Tube. 
The cathode ray is projected at P parallel to the axis of x, and the electro- 
static deflection is produced by the charged plates A and B, which are kept 
at a constant potential difference. The dimensions of the tube are shown in 
the figure, the distance between the plates being 0*69 cm. 
The stray electrostatic field between P and 0^ deflects the ray before it 
comes between the plates. A small deflection at 0| would produce a con- 
siderable deflection at the distance Q, even supposing there was no further 
deflection by the electrostatic field. 
Let us first consider the effect of the stray field between P and 0^. 
