EQUATIONS OF PROPAGATION OF ELECTRIC WAVES. 
41 
at both boundaries. Taking k and k to be two possible values of k, and X and X' the 
corresponding’ forms of ^„{Kr), we find 
— fc"~) XX'(Ir = 
dx' , dx 
X — — X —- 
dr dr 
and then, Ijy the usual limiting process,"^ 
3 1 
X\lr = - 
or 
'dx dx ^ d dx' 
die dr die dr ’ 
_ + 1) I 2«+3 ; / \ ^ 3 
^ ^2^.2 I'U (X"W'0O 
/C y T ' 
Wlien i\ is very nearly equal to Vq, this becomes, approximately, 
I = ('o —+ i)’A<x»('"'o)}". . 
(91). 
After the appropriate expression (90) or (91) has been substituted in the expression 
(88), the total energy, kinetic and })otential, is to be obtained by suppressing the 
factor sin® k Gt. Thus, when i\ is very nearly equal to r,,, the total energy of the 
oscillating charges on the condenser is 
. n~ {n + 1)“ 
’4 2« + r 
'.)V 
(92). 
Communication oj Electrical Oscillations to External Medium. 
34. AVhen there is no aperture in the outer conductor, tlie oscillations considered 
in §§ 31-33 would, in the absence of dissipation due to imperfect conduction, continue 
indefinitely ; but they would not produce any effect in any external electrical system. 
When there is an aperture, we may take account of it by supposing that tlie displace¬ 
ments (magnetic and electric), in space external to the condenser, are of the character 
corresponding to waves diverging from sources distributed over the aperture only, 
and that the displacements within the dielectric plate of the condenser difier from 
those, which would be found in a normal mode of oscillation, by the supeiqiosition of 
dis})lacements corresponding to a system of standing waves, which are insensible 
except in the immediate neighbourhood of the aperture. We may suppose the 
* Cf. Lord Rayleigh, ‘ Theory of Sound,’ vol. 1, p. 325. 
VOL. CXCVII.-A. 
Gr 
