668 Prof. W. McF. Orr on the Impossihillty of 



damping^ by evaluating the radiation across the tubular 

 surface generated by a small sphere of radius p whose centre 

 is moved round the circuit. He states that the electric force 

 at any point on this surface consists of a part which varies 

 as Ijp, together with a part which involves p as a factor since 

 it must vanish with p. One is, however, justified in stating 

 only that the ratio of the second part to the first must 

 vanish with p ; a similar remark applies to his expression for 

 the magnetic force. The proof is consequently invalid : such 

 a line of argument can in fact legitimately show- only that 

 -as the wire becomes infinitely thin the rate of radiation be- 

 comes indefinitely small compared with the logarithm of the 

 ratio of the wave-length to the thickness of the wire. 



4. Furthermore, it may be shown directly that any of the 

 solutions given by Macdonald involves radiation, and that 

 this remains true if we combine two such waves of the same 

 period travelling in opposite directions round the ring. 

 Whether we adopt Poynting's expression for the rate of 

 radiation across a closed surface or Macdonald's modification 

 of itf, the mean rate for a complete period^ supposing that 

 there is no damping (energy being supplied to the conductor 

 if necessary)!, is equal to the mean time value of 



where a, y8, y denote the magnetic force, X, Y, Z the electric 

 force, and /, ?7i, n the direction cosines of the outward drawn 

 normal, the integral being taken over the surface. 



If we have any system of perfect conductors performing a 

 simple harmonic vibration, regarded as undamped (energy 

 being supplied to the conductors if necessary), and if S be a 

 sphere of radius R, large compared with the dimensions of 

 the conductors, enclosing them all, and such that the distances 

 of its centre from them is comparable with their dimensions, 

 the magnetic and electric forces at points on S will each 

 contain, prima facie at least, terms of order 1/E. In such a 

 case then w^e require in the above integral to take account of 

 terms of this order only. To this order the electric and 

 magnetic forces are at right angles to each other and to the 

 radius of the sphere, and the integral assumes the form 



where Y is the velocity of transmission through the medium. 

 Accordingly, any system whatever executing harmonic -sdbra- 

 tions of one definite period wdll radiate energy unless, to the 

 order 1/R, we have a = 0, /3=0, 7 = 0, at all points of the 



* Loc. cit. p. 83. t Loc. cit. p. 72. 



X The discussion is simplified by thus ensuring that the forces do not 

 involve the time exponentially. 



