EQUATIONS OF PEOPAGATION OF ELECTEIC WAVES. 
45 
and, at any rate when n is not too great, this is a very small fraction, if is 
large compared with a, as has been supposed throughout the investigation. 
37. The form of the result shows that the number of vibrations of the higher- 
modes, that are executed before the disturbance sinks into insignificance, is much less 
than that of the lower ones. The occurrence of ry — 7 ’^ in the denominator of (103) 
suggests that the principal factor in securing permanence of the vibrations is not the 
capacity of the vibrating system, but the screening action of the external conductor. 
The latter point might be illustrated further by considering the example of a spherical 
condenser, in the case where 75 is small compared with Tq. The boundary condition 
at the inner surface can be satisfied approximately by putting, in ecpiation (83), 
B = 0 ; and the frequency is determined by the equation 
= 0 .(104), 
when r = jq. The total energy, for a mode of oscillation given by o),, = r" P„(/j.), is 
i j 1 _ + 1 ) 1 2«+3 r,/, \ m 
^ 2/1 + 1 ^ J ^ xPA'^'oJf > ■ ■ 
and the energy dissipated in a. period is 
KV(n + 1)- I + +K‘r„") ; 
. (105), 
■ (io«) 
and it is clear that the fraction of the total energy dissipated in one })eriod is of the 
same order of magnitude as before, except that («/>q)^ is suljstituted for the product 
of and {a/(/’o — '>\)]- For the mode of least frequency n = 1, and we have 
kTq = 2'75 nearly,'^ instead of 1'41, its value when is nearly equal to (q ; and thus 
the fraction in question becomes apj[)roximately 
! a 
a \8 
8277 ( ) > instead of being approximately 1277 fy- ) —^ ; 
or the rate of dissipation of energy, for the spherical condenser, is less when tlie 
capacity is very small than Avhen it is very large. 
* See J. J. TnoMSOX, ‘Eecent Eosearclios,’ p. .373. 
