3840 Radiation from Alternating Circular Electric Current. 
and by doing so it becomes 
(aKa) a f1—(—)? sin (2«a sin @)'dé 
0 
— (2x) ap 
Thus the rate of radiation becomes 72C2caV/2, but, as 
before, this expression is to be doubled in case c=0. 
7. Any periodically alternating current can be resolved 
into a number (in general doubly infinite) of elementary 
currents each of the simple harmonic type discussed aboye, 
and it is evident that the mean rate of radiation is the sum 
of the mean rates of radiation due to each of such constituents 
separately. 
It appears from this and from the result of § 6 that if the 
current is a simple harmonic function of the time, and if the 
wave-length in free space is small compared with the radius, 
the radiation approximately depends only on the mean value 
of the square of the current averaged round the circle, and is 
otherwise independent of the law of variation from point to 
point. 
8. Lord Rayleigh, in an investigation of the work done by 
given forces applied at given points of an elastic solid, has 
referred * to the case when the forces act tangentially along 
a circle, in connexion with the subject of the present note. 
He states that ‘‘it would seem that (33) must lead to a more 
complicated expression for the energy radiated than that in 
Dr. Pocklington’s investigation.” As I understand it he 
considers in his expression (33) the applied forces IF’, F'’, to 
be the analogues of electromotive forces and proposes to 
replace them therein by expressions in terms of the conduction 
current obtained from Pocklington’s resuits. The parallelism 
between the electromagnetic and the elastic-solid theories 
does not appear, however, to extend so far. A current of 
conduction presents itself in the electromagnetic equations as 
a discontinuity in the time-rate of change of the electric 
forcet. The force F applied to the elastic solid is the 
analogue, not- of an applied electromotive force, but of 
—du/dt, where uw is an impressed conduction current; 
accordingly in Lord Rayleigh’s expressions for the displace - 
ment of the medium at a point P due to a force F applied at 
another point O, F is ‘to be regarded as the equivalent of 
—du/dt. Yet again, in obtaining the rate of radiation we do - 
not multiply F’, or —du’/dt, by the velocity of displacement 
* Phil. Mag. Oct. 1903. 
+ Compare Macdonald, ‘ Electric Waves,’ p. 16. 
