322 
PROF. H. M. MACDONALD ON THE EFFECT PRODUCED P.Y 
edge, and let QP make an angle 9 with the line QPt through Q in the screen 
perpendicular to the edge, then 
K' = 
1) 
where r now denotes the distance of P from a point in the line QPt; for points near 
to Q 
9-^ = 2riP cos 9, 
where P is the distance QP, and therefore the principal part of K' is 
ikH I* CP jn tKl\ COS 3- 
P 
Kli |*CC 
k Jo 
(h\, 
that is, K' is a quantity of the order (kP)“‘^- ; hence at points which are at a distance 
from the nearest edge great compared to a wave-length the effect of the edges is 
negligible in comparison with the waves due to the oscillator. Therefore the principal 
parts of the components of the electric and magnetic forces at such points are equal 
to the princi})al parts of the electric and magnetic forces due to the assumed electric 
current distribution and the oscillator. 
It may be verified that this agrees with the known solution of the problem of the 
semi-infinite conducting plane. Taking the case where the electric force in the 
incident waves is parallel to the edge, let the origin be in the edge, the axis of 2 
along the edge, the axis of y perpendicular to the plane of the edge, and let the 
direction of the incident waves make an angle Avith tlie conducting plane, then 
the conqjonents of the electric force in the incident waves are 
X = Y = 0, 
Z = 
gi(c{Vi+j:cos sill 5,) 
and the components of the assumed electric current distribution on the upper face of 
the plane are 
a = 0, = 0, ” ^ 
and on the lower face 
a = 0, ^ — 0, y = 0. 
Hence the components of the electric force due to the assumed distribution are 
X = 0, Y = 0, Z = — (27r)“h/C sin -li [ [ (3‘''(Vi + x,cos3,-r) ^,-i 
where (.C], 0, z^) is any point on the screen, r is the distance ot the point P (a-, y, 0) 
from the point on the screen, and the integration is taken over the conducting plane. 
Noav 
9-3 = (x-a-if+ ?/' + 2T = p" + 2Y, 
