EQUATIONS OF PROPAGATION OF ELECTRIC AVAVES. 
35 
We write 
dxdz 
d dr ^ 
dx' dz ’ 
and similarly for y, and integrate by parts ; the result 
contains a line integral round the boundary of the aperture and a surface integral, 
and the former contributes nothing to the limit we are seeking, unless the point Q is 
indefinitely close to the boundary of the aperture. Thus the limit we are seeking is 
that of 
dz /-\dx' 
K'v^) COS K (ct — r) — Sk)' sin k (cl — r) [ 
kV^) cos kc + 3fcr sin k/'} cos kcI 
—[ — K^r" sin k(cI — r) + K^r cos K(ct — r) 
[K^r'^sm KV + kVcos kv] cos k cl ; 
s 
in this expression, the two last lines vanish in the limit, and the others yield the value 
at Q of 
. _3 /d.q^ 9/d, . 
Thus we have 
477 lim (P) — (P') ] = 477 (Q) cos K ; .... (95)^ 
and it follows that the conditions of continuity are satisfied by putting 
il l = ^ 
.A — Ji ’ J2 — ’ 
(76). 
By this result, the transmitted waves on the further side are connected with the 
waves on the nearer side; and it is manifest that the result would not be disturbed if 
the surface were not plane, provided that all the linear dimensions of the aperture are 
small compared with the radii of curvature of the surface. 
[30. (Parihj re-wrilten March, 1901.)—-We return now to the general question 
propounded in § 24, and seek to estimate the character of the answer that we haA^e 
found. In § 25 the question is made more precise by showing that the distribution 
over the aperture of sources to which the transmitted radiation can be regarded as 
due, depends upon the values, at certain times and at points within the aperture, of a 
certain system of magnetic and electric displacements ; these values are the quantities 
denoted by u^, . . . In § 26 the transmitted radiation is expressed in terms of these 
2 F 
