Light by a Dielectric Cylinder. 369 



work for a transparent sphere *, so that k'/k=l'5. And 

 before employing the more general formulae, I commence 

 with the approximations of (6) and (9), assuming kc='10, 

 k / c = '15. When we introduce these values into (6), we get 



a/t=4= /"-^- N )V^-^r«00625--156xlO- 4 cos6>], . (13) 

 T K \2ikr/ L 



in response to the incident wave h/K — e i{nf+kx) . Again, 

 from (9) 



yjr = c= rJjL)V«*-*> [10- 4 ('0781 + -0481 cos 26) 



-■00385 cos 0], .... (14) 



corresponding with c = e i( - nt+kx) for the incident wave. 



In using the general formulae the next step is to express 

 yfr m , representing a divergent wave, by means of functions 

 already tabulated. I am indebted to Prof. Nicholson for 

 valuable information under this head. It appears that we 

 may take 



yjr m (z)=G m {z)— iiirJ m (z), .... (15) 



where z is written for kr, and the real and imaginary parts 

 are separated. When z is very great 



i'"*4z) = (£fe-" (16) 



J m (z) is the usual Bessel's function ; the G-f unctions are 

 tabulated in Brit. Assoc. Reports |. The Bessel's functions 

 satisfy the relations 



T 2m ' 



«»i+l = J»-Wi«-1) .... (17) 



J»i — Jwi-i ;J»m .... (18) 



and relations of the same form are satisfied by functions Gr. 

 When m=0, Jq^-J^ G^-G^ 



Writing z for kc and z for k'c and with use of (18), we 

 have for the coefficient D m of *2i m on the right-hand side 

 of (5) 



D m =/J^)J m _ 1 ( lS ')~^Jm(^)J»*-i(^); • • (19) 

 and for the coefficient of B TO on the left 

 N OT + ^7rD OT , 



* Proc. Roy. Soc. A, vol. lxxxiv. p. 25 (1910) ; Sci. Papers, vol. v. 

 p. 547. 



t Reports for 1913, p. 30 ; 1914, p. 9. 



