2!) 
EQUATIONS OF PROPAGATION 
OF ELECTRIC WAVES. 
being regarded as due to iiiiagiiied sources, situated in tlie aperi-ure, can be calcu¬ 
lated directly from the formulae of ^ 1 G, by fii’st assigning certain functions of x', 
y', t as the forms of'W, . . . under the sign of integration, then suhstituting t — r/c- 
for C and finally integrating over the aperture. We shall take the forms, that 
are to he substituted, for u, . . . under the sign of integration, to be given by the 
equations 
II = cos KCt Mo sin K ct, 1 
j 
f — /-j cos K Qt + sill K ct , 
. ( 6 ( 1 ), 
where ili, 
are functions of x', y', for which 
^ “ 8 / ’ 
( 61 ). 
and similarly for Further, we shall denote the values of n, . . . , resulting from 
the integrations, by u+, . . . The answer to the general question of § 24, will thus 
lie in the determination of the functions ii^, . . . These functions can be regarded as 
the values, at certain times, and at points within the aperture, of a certain system of 
magnetic and electric disjilacements.] 
26. Before proceeding it will be convenient to record the forms for u^, ... in 
terms of the functions u^, . . . It will lie sufficient to put down the terms that 
contain cosKCt. We ohsei've tliat in the formulse of § 16 
.7f — .r 
or ^ 8r ^ 
/ ) 
O./' o.r 
2 _ ( r - x'f + (?/ - y'f _ 8-r 1 
,,5 
o(?/ - ?/')(- - 8 V -1 
8 b'-i 
v> 
djjdz 8 //' 8 .:' ’ 
and we also observe that, when tlie surface S is a portion of a plane {z = z), we must 
have / = 0, m = 0 , m = 1 , tlie point {x, y, z) lieing on the side s > z. We can 
therefore write down tlie formiilie for m+, . . ./’+> • • • follows :— 
(Jx'd y' 
— Mj {cos K (cf — r) — Ki'-HinKict — r)] 
I 
q7i ( cos k{cI — r) — Kf sin k {ct — r) 1 
8 b--i 
8 .r" 
+ i 
0-1 
.-1 
ar 8 
xy/ 
■“ K')-') cos K {cl — r) — 3A:rsinK(t2 — r)' 
(?/ “ y'f + G' — , 
- -;;- !/,>,cosk{ci - r)} 
■ (63), 
