SCATTERING OF PLANE ELECTEIC WAVES BY SPHERES. 183 



Similarly, we write 



(3'5) E,(z) = 2 ".(-i|)'( 



\ z dz! \ z 



-a ()-*,, (*x 



where 



2(l-2n) 2.4(l-2n)(3-2) 



In terms of the K B function (the modified Bessel function used by MACDONALD) 

 we have the relation 



(3-51) 1 



Thus 



E_() = v-iu 



in terms of the notation used by MACDONALD in the paper last quoted, and in 

 LAMB'S notation 



(37) E B (z) = z" +1 {* n (z)-,+ n (z)} = z n+i f n (z). 



In consequence of the equation ("2'6) we see that both S n (z) and E n (z) are solutions 

 of the differential equation 



(3-8) 



The functions S n (z), C n (z) and E B (z)| have been tabulated from z = 1 to 10, and 

 for values of n ranging from to 22, by Mr. DOODSON,* and these tables have formed 

 the basis of the numerical calculations mentioned on p. 176 above, t 



It will be convenient to collect here the simple relations amongst the functions 

 S B _i, S B , S B+1 , which correspond to the known results for Bessel functions, or to those 

 given by LAMB for the equivalent function !/< (z). 



Difference delations for the Functions S n , E n . 

 From (3 4) we see that 



(3-81) S, l+1 (,) = - 



and by using (3 5) we see that the same relation holds for E n (z). 

 Again, it will.be found that 



/A+ 

 \az 



* 'British Association Report,' 1914. 



t PROUDMAN, DOODSON and KENNEDY, 'Phil. Trans. Roy. Soc.,' A, vol. 217, 1917, p. 279. 



VOL. CCXX. A. 2 D 



