166 Dr. L. Silberstein on Radiation 



the sphere =1. Then the electromagnetic field within the 

 sphere is, by the equations (b), 



K3E/3* = c . curl M, dM/^= c . curl (e-E) *, 



divE = divM = 0, 



t> • a( n **i. 1 ^\ p 2 cos 30 



F = sin 6 ( -g <£ -f- ^r- ^~ ) , R — — — ^ 

 \c 2 T Kr or J Kr or 



,, sin # B 2 c£ , oA / . , A 1 • nr' 



]\/[ — _r_ . 6 = — 2 A cos (nt + rj).~ sm — » 



c ^r^t r r v J 



^ (6) 



where u=cK~ 1/2 is, in the case of a real positive K, the 

 velocity of propagation of disturbances within the sphere ; 



c 2 

 more generally, K ■== -^ (1 — ixt) 2 , where cr is the " extinction 



index" and c = y/ — l ; then K in the first differential equa- 

 tion stands for a differential operator, as in the usual 

 treatment. Except in the immediate neighbourhood of 

 a convergence point the influence of a in our dispersion 

 formulae will be disregarded. Our K will therefore in 

 practical applications be always real, and generally posi- 

 tive ; but it may, in certain regions of the spectrum, acquire 



c 2 

 negative values ; then K = — 2 (1 — o- 2 ) and a > 1 ; such 



regions will, in fact, be especially interesting in connexion 

 with band spectra. 



The meaning of the remaining symbols in (6) is as 

 follows : — R the radial, and P the meridional components 

 of the electric force E ; M the intensity of the magnetic 

 force, the magnetic lines being circles of latitude, on each 

 sphere concentric with a ; axis coinciding with e ; 6 angle 

 contained between axis and any radius vector ; finally, A 

 and v functions of the frequency n, which, together with 

 the corresponding magnitudes A', i/, outside the sphere, are 

 determined by the boundary conditions 



(KR) = (R'), (P + «Bin^ = (P'), (M) = (M'), . (6«) 



where () means "for ?* = a," and dashed letters refer to the 



* The impressed force being homogeneous, curl e vanishes everywhere 

 (leaving only a vortex sheet on the surface of the sphere), bat the presence 

 of e is essential in as far as it leads to the boundary condition : tangential 

 component of E — e (not E alone) continuous. The remaining condi- 

 tions are as in the usual Maxwellian theory. These boundary conditions 

 determine A, n in (6) and the corresponding A', »;' in the held outside 

 the sphere. 



