Light by a Dielectric Cylinder. 



367 



suppose he and k'c to be small ; and we find approximately 

 for the secondary disturbance corresponding to (3) 



H0 eKnt '^ 



p^(k- 



Pc 2 ) 



8 



cos 







(6) 



showing, as was to be expected, that the leading term is 

 independent of 0. 9 ' 



" For case 2, which is of greater interest, we have (from 

 the general equations) 



(*i* + *-i* + i) e = . . . . (7) * 



\dxPdx dyPdy ) ' 



This is of the same form as (2) within a uniform medium, 

 but gives a different boundary condition at a surface of 

 transition. In both cases the function itself is to be con- 

 tinuous ; but in that with which we are now concerned the 

 second condition requires the continuity of the differential 

 coefficient after division by P. The equation for B OT (or B„/ 

 as we may write it for distinctiveness) is therefore 



B m '[k' 



df m j 



d.kc 



dJjk'c)\ 

 d./e'e ) 



(k'c) -kef ri 



=2i m {kc^ m (kc)JJ(k'c)-k'cJ^k'c)J m '(kc)}, . . (8) 



with the understanding that the 2 is to be omitted when 

 m = 0. Corresponding to the primary wave e iint+kx \ we find 

 as the (approximate) expression of the secondary wave at a 

 great distance from the cylinder, 



IT v 



Hmr) ei 



_fLl (k*c 2 -M 2 c*) 



-I 



-V.2 



P 2 -P 

 k" 2 + P 



cos 







8 P + P 2 



cos 2(9 



]• 



(9) 



The term in cos 6 is now the leading term ; so that the 

 secondary disturbance approximately vanishes in the direc- 

 tion of the primary electrical displacements, agreeably with 

 what has been proved before. It should be stated here 

 that (9) is not complete to the order Pc 4 " in the terms con- 

 taining cos 6. The calculation of the part omitted is some- 

 what tedious in general ; but if we introduce the supposition 

 that the difference between k" 2 and P is small, its effect is to 

 bring in the factor (1-JifeV) " 



* In (7) c is the magnetic component, and not the radius of the 

 cylinder. So many letters are employed in the electromagnetic theory, 

 that it is difficult to hit upon a satisfactory notation. 



202 



