42 
PROFESSOR A. E. IL LOVE ON THE INTEGRATION OF THE 
fliverging waves, and tlie standing waves, thus introduced, to have the same period 
as the oscillations in the normal mode ; and then the displacements corresponding 
to them will l)e determined, as in § 29, Ijy the conditions that the electric and 
magnetic displacements must be continuous across the aperture. As we are con¬ 
cerned rattier with the c’eneral features of the transmission of disturbances across an 
O 
aperture tlian with sjiecial details, we may select any normal mode of oscillation for 
examination. We shall supjiose the aperture to lie a circle of radius «, small com- 
jiared with the distance between the conducting surfaces ; and we shall con- 
sidei' particularly modes of oscillation symmetrical about the axis of the circle, taken 
as axis of 2 . 
35. For the calculation of tlie energy dissipated we shall take, in the notation 
of § 25, 
/ - - - \ n (71 + 1) , , , , , , 
(wi, . (y, - a;, 0) 
= ^ • (?/, “ < 0), «ay,.(93), 
where ——- lias l)een written for ; and we shall take 
2 \ lA = 1 
(./n ^i) = - « 0^ + y'^ ’o) 
= 1 y\ ^’o), say,. 
these lieing with sufficient approximation the values obtained in § 32 ; the normal 
mode of oscillation here discussed will accordingly he one for whicli the axis 2 is an 
axis of symmetry. 
We have now to find the most important terms in 477?^ + , ... at a great distance 
from the apei'ture, the values above written l)eing substituted for ?7.j, .... A e 
shall take U for tlie distance of tlie point {pc, y, z) from the centre of the aperture, 
and, whenever we wish to do so, w^e shall expand r in the form 
r 
— 11 fi- 
X ' 
i; 
+ y' 
R 
+ 
Now, taking imi ^, the first line of equation (G3) is 
y 
I -|- 3 
,ra; + yy 
R2 
[sin K (f’': — R) (— kR) 
+ cos K — R) [1 —K~ {xx 
+ !/?/')}] 
approximately, where the integration is lakeu over the area within the circle 
x''^ = rr, and terms of order higher than x'^ have been neglected. The most 
im])ortant ])art of this is 
