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120                  ELECTRICAL VIBRATIONS  ON  A   THIN  ANCHOR-RING                [365
The calculations just effected afford an independent estimate of the dissipation. The rate at which energy is propagated outwards away from the sphere of great radius r, is
dE     rr ,      -- =- = V. 47rr2 .
.         .                       -
at                      T-     in
or, since  T  (the  period) = Z-jra/mV, the loss  of energy in  one complete vibration is given by
dE.r    87r4aW^             T
t' 2771+1
dt             ni-
With this we have to compare the total energy to be found within the sphere. The occurrence of the factor e2^'' is a complication from which we may emancipate ourselves by choosing r great in comparison with a, but still small enough to justify the omission of e*a*r, conditions which are reconcilable when e is sufficiently small. The mean value of P2 + Q2 + R2 at a small distance p from the circular axis is 2m2/azpz. This is to be multiplied by ^Tra.^TTpdp, and integrated from e to a value of p comparable with a, which need not be further specified. Thus
    SmV2 (dp       SraV8.                                 ,^^^
E =----    - =---------loge;    ..................(50)
a    j  p           a                                    ^     '
dE.r_ TT- {Jam-i (2m) - Jnn+i(2w)}
in agreement with (23).ance ; but, as Lamb has shown J, it is to be expected in such cases as the present, where the vibrations to be found at any time at a greater distance correspond to an earlier vibration at the nucleus.