670 Prof. W. McF. Orr on the Impossibility of 



It may be shown that Fi(^)±F2(^) = according as a is 

 even or odd. 



The coefficient of Ci in y3 



= ?VE"' cos eJ'^"^-"^^ I sin ^{r . ^i^(^+"n0sin^)^^^ 



n + TT 

 '"^ I sin yjr sin <7(i|r + sin 6 sin -^)dyfr, 



= -crR'"'cos^.^' 



+ 7r 



iK(a(p-R) r 



= -0-R-' cos eJ'^'"^''^^Me), 



and the coefficient of C2 



=-0-11"^ cos ^.e'''^-"^-^) r ''sin i/r sin <T{-^fr + sin ^ sin yjrjdyjt, 



^-aU-' cos eJ''^-'"^-^^f^{d). 



It may be shown that/i(^) ±f2{6) =0 according as cr is odd 

 or even. 



The coefficient of Cj in 7 = o-R-' sin OJ'^'^-^^Me), 



and that of C2 = o-R"' sin OJ^^-^^-^^id), 



Inserting the time factor and selecting the real parts, we 

 have 



«= -o-R-^ cos <9{CiFi(6>) sin (a0 + V^~R) + CsFsd?) sin (-«(/) + V^-R) }, 



fi = -o-R-^cos l9iCi/i(6>) cos (a0 + Vi5-R) + 02/2(6^) cos (-a^ + Y^-R) }, 



7 = o-R-^ sin e{Cifi{e) cos (a^ + Y^-R) + C2/2((9) cos (-a<^ + Y^-R)}. 



It is evidently impossible that any one of these components, 

 far less all three, should vanish for all values of 6, <^, t. 

 There must then be radiation. 



5. This result may be extended ; it may be proved that per- 

 manent free oscillations in any indefinitely extended dielectric 

 space are impossible. For let us suppose that the vibrations 

 are undamped, being maintained by energy supplied to the 

 conductors if, as is here held, this be necessary, and describe 

 any sphere inclosing the latter. Outside this sphere the 

 components of the vector-potential F, G, H can be written in 

 forms of the type 



F=:2^En(/^9').F„,,.^-^^ 



where E^(p) denotes (dlpdpY{p-h-'^), and F^,^, G„,«, H^,,;^ are 

 any solid harmonics of degree n, the summation being extended 

 over all positive integral values of n, including zero, and 

 over values of k corresponding to the periods of all the con- 

 stituent simple vibrations *. The components of magnetic 



* See J. J. Thomson, Proc. Lond. Math. Soc. vol. xv. 'Recent Re- 

 searches in Electricity and. Magnetism/ pp. 361 et seq. 



