Electromagnetic Theory of Light. 99 



function and its differential coefficient with respect to r shall 

 be continuous. The divergent wave is expressed by 



Bq^o + B^ cos 6 + B 2 ^ 2 cos 20 + ... , 



where ^ , i^, &c. are the functions of kr defined in § 341. 

 The coefficients B are determined in accordance with 



B - { kG ^ Jm(¥c) ^ c +- TJfc J «W } 



= 2i m { Vc J m {he) J m '(Vo) - he J m (k'c) J m f (kc) } , 



except in the case of m = 0, when 2i m on the right-hand 

 side is to be replaced by i m . In working out the result we 

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

 the secondary disturbance corresponding to (64) 



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

 pendent of 9. 



For case 2, which is of greater interest, we have from (15), 



(fM*rM +f ) e=0 - ■ ■ (66) * 



This is of the same form as (63) within a uniform medium, 

 but gives a different boundary condition at a surface of transi- 

 tion. In both cases the function itself is to be continuous; 

 but in that with which we are now concerned the second con- 

 dition requires the continuity of the differential coefficient 

 after division by k 2 . The equation for B TO is therefore 



B fjc'c^J (k'c)-kc<r d MM\ 



= 2i m {kc J m (kc) Z m '(k!c) — k'c J m (k f c) d m '(kc)}, 



with the understanding that the 2 is to be omitted when m = 0. 

 Corresponding to the primary wave e^ nt+Jcx) , we find as the 

 expression of the secondary at a great distance from the cy- 

 linder, 



-WJ^o«*-Jw{^™,M]. .(67) 



The term in cos is now the leading term; so that the second- 



* In (66) c is the magnetic component, and not the radius of the cylin- 

 der. So many letters are employed in the electromagnetic theory, that it 

 is difficult to hit upon a satisfactory notation. 



H2 



