ATMOSPHERIC ABSORPTION AND SCATTERING 



155 



real frequency of constraint of the radiation field 

 falling on a sphere, as in the jDresent case. The de- 

 nominators of the amplitudes a,, and &„, although re- 

 duced, can never become zero, and there are no diffi- 

 culties caused by resonance. 



A glance at the formulas (14) shows the com- 

 plexity of the amplitudes a„ and b„. An exact com- 

 putation of these coefficients is out of the question on 

 account of the lack of tables of Bessel and Hankel 

 functions of complex argument in the range needed 

 here. They reduce to simple expressions in tlie limit 

 when the parameter p = 2wa/\<^ 1. In the present 

 work we shall be mostly interested in the cases where 

 p < 1 or /D ^ 1. In these cases a series expansion of 

 the anri)litudes in ascending powers of the parameter 

 p can be used. With the expansions of the spherical 

 Bessel and Hankel functions 



^J"(p) = 2"p'.Z (-)-(- + -)' 



ro?w!(2?i + 2m-|- 1)! 



:„«(p) = 2" p" X 



( — ) "" ( n + ffO ! 

 'g m\{2n-\-2m + 1)! 



+ 



E 



(2n-2m)! ^,^ 



22 -Ti + i ^—' in\ in— m )! 



used in equation (14), one is lead to the following 

 amplitudes, keeping the first few terms of the ex- 

 pansions and assuming jUj = p.„. i.e., the equality of 

 the permeal)ilities of the medium and the sphere: 



L \2n + 1 2(2«-t-5)/ J ' 



K^-jr-"( ''■ Y 



■' \{27l+l)\J 



(2n+l)(n-f 1)(A^^-1) ^^^^ ^ 



(36) 



[ 



1 + p' 



nN^ + n+1 '^ 



(2/1+1) [(2w-l)A^^- «. -1] 

 (2/1 -t- 3) (2/1- 1) {nm^- 11+ 1) 



\{2n + 1)!/ 



(2/;. + 1) (// + 1) (A^^ - 1) 2„-i-i^ 



(37) 



From these expressions one derives at once the ex- 

 plicit formulas representing the induced magnetic 

 dipole (aj, electric dipole (&i), and electric quad- 

 rupole {h.,) amplitudes. One has, then, neglecting 

 powers of p higher than the sixth, 



fli = 



45 



{m- l)p\ 



6i 



-2jN^-- 1 

 3 A^ + 2 



2, N--2 



5 iV2 -f 2 



2j N' - 1 ,\ 

 3 N-^ + 2^ )' 



(38)' 



and 



_ - .;■ N^-l , 

 15 2A2-f 3 



It would appear interesting to present the relation- 

 shij)s connecting the amplitudes of the electric and 

 magnetic poles a„ and 6„ with those appearing in the 

 treatment of Mie which was used by Eyde." The 

 magnetic and electric amplitudes in Mie's notation 

 are respectively pn and a„, and the relationships in 

 question are the following: 



p^Mie= {-Y j {2n + I) an, 



«„«''= (-)"+'i (2/1+1) 5„. (39) 



Finally the formulas (38) can l>e transformed so 

 as to have the real and imaginary parts of tlie ampli- 

 tudes separated easily. The refractive index N of the 

 spheres is connected with their complex dielectric 

 constant by 



ic = N^ = ir — ja, 



or with 



AT = n (1 - jx), 

 tlie complex index of refraction, one has 



(40) 

 (41) 



(42) 



'Some misprints and slight errors in the expressions for 

 these amplitudes may be noted in reference 18a. On page 571 

 in the formula (35) and in the denominator of the coefficient 

 of p2, read (2/j + 2) instead of (2/i. + 1). In the formula (36) 

 the minus sign on the right-hand side is missing. In reference 

 18a, bi'' and 62'' have the wrong sign and the p' term in fci' 

 is incomplete. It is recalled that +i has been replaced through- 

 out this report by —j. 



