32(3 
PEOF. H. M. MACDONALD ON THE EFFECT PEODUCED BY 
As a fuither example, tlie case of an Infinite plane conducting screen with a circular 
aperture of radius a will be worked out. As above, let a straight line through the 
centre of the aperture perpendicular to the plane of the screen be the axis of ?/, the 
axes of X and 2 l^eing in the plane of the screen, and let the incident waves be 
given by 
« = 0, = 0, y = 
the magnetic force being parallel to the plane of the screen and the direction of the 
waves perpeiidicular to it; then the components of the magnetic force due to the 
distribution to be assumed over the screen are 
= 0, = — e 
iTT 
r Pi 
4- — dS, 
J cz r 
yo 
277 
1 a 
JJ Py 
r 
f/S, 
where r is the distance of the point from a point in the plane of tlie screen and the 
integrals are taken over the conducting part of it. Transforming to cylindrical 
co-ordinates 
r- = + cos (</>!-</)), 
where the co-ordinates of the point P are (p, (/>, y) and of a point in the plane 
(pi. 0)- The principal parts of the integrals are contributed by the elements in 
the neighbourhood of the point for which r is stationary, that is, when 
</>i = pi = p, 
and then, provided p is not small. 
® f/S = ‘ ‘ ^ 
Pi f/pi c/(/)i = 
a J 0 / \ ^ 
-Vs 
where = y‘‘+ {pi + pf. 
For a point on the negative side of the screen y = —d, and writing pi = p + p^ the 
principal part of the integral j*!* dS is given by 
J a —p 
that is, provided /cp^ is great compared with d, by 
J 
where Uq = 2’''“ (XcZ)“”’(a—p). 
Hence, at a point P on the negative side of screen, the principal parts of the 
