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Full text of "Scientific Papers - Vi"

1918]       ON THE  DISPERSAL OF  LIGHT BY  A  DIELECTRIC  CYLINDER         557
with (8) after correction of some misprints. His function Q corresponds with my >/r, at least when we observe that the introduction of a constant multiplier, even if a function of m, does not influence the final result.
In proceeding to numerical calculations we must choose a refractive index. I take for this index 1*5, as in similar work for a transparent sphere*, so that k'/k = I'D. 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
e*(nt~*r) ['00625 -- -156 xlO-4 cos 0],   ......
in response to the incident wave hfK=ei(nt+kx).    Again, from (9)
ty = c = (-}  e*<nt-fei [1C-4 ('0781 + -0481 cos 20) - '00385 cos 0],...(14)
\a1tlCTj
corresponding with c  eiint+kx} for the incident wave.
In using the general formulas the next step is to express ^rm, 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
tym(z)*= Qm(z)~ \i-rr Jm(z\  ..................... (15)
where z is written for kr, and the real and imaginary parts are separated. When z is very great
*(*)-*ll(ss)' e~iz ......................... (16)t
Jm(z) is the usual Bessel's function; the (-^-functions are tabulated in Brit. Assoc. Reports j. The Bessel's functions satisfy the relations
"m+i ** ~~~ "m ~ "m i >    ........................ (17)
Z
Jm = Jm-\ ~~ ~~ "m > ........................... (18)
z
and relations of the same form are satisfied by functions G.    When m = 0, i/o =  t/1} (TO =  Gri.
Writing z for kc and z for k'c and with use of (IS), we have for the coefficient Dm of 2im on the right-hand side of (5)
Dm = *'Jm (z) Jmr-i (*') ~ zJm (z) Jnr.1 (z)\ ............... (19)
and for the coefficient of Bm on the left
* Proc. Roy. Soc. A, Vol. LXXXIV. p. 25 (1910) ; Scientific Papers, Vol. v. p. 547.   * [t A correction has been made in this formula.   In order to yield (15), ^OT (2) must be deduced from i//0 (z) by (5) of  341 of Theory of Sound, and not by (11) of ihat section, which gives a different law of signs ; also ^0 (z) must be taken with the opposite sign from (18) of  341.   W. P. S.] J Reports for 1913, p. 115; 1914, p. 75. also agreesion.again, as in (1), so that the vibration is A sin2 6 + C cos2 6, reducing to G simply, if A = G. This is the result for a single particle whose axis is at W. What we are aiming