96 



LORD RAYLEIGH ON THE ACOUSTIC SHADOW OF A SPHERE, 



The analytical solution is expressed in (17), which we may compare with (15). 

 Restricting ourselves for the present to the factors under the sign of summation, we 

 see that the coefficient of P in (17) is 



__ (2_+ l)c'^x,, (c) = (2n_+ 1) c 



while the corresponding coefficient in (15) is 



(2n + l)i" 

 "D,, 



If these coefficients be called C,,, C',, respectively, we have 



C. = i--o" +1 x.(o).d' (20), 



in which the complex factors ti+\ u (c), C',,, for c 10, have already been tabulated. 

 We find 



The product above tabulated shows marked signs of approaching the limit 2, as 

 n increases; so that the series (17) is divergent when P = 1, i.e., when 6 = 0, as 

 was of course to be expected. The interpretation may be followed further. By the 

 definition of P,,, we have 



f -t fy ft i^ O 



(. * * I 



so that, if we put a = 1, 



1 + P! + P 2 + P 3 + . . . = 



= 1 + P, . a + P.. a~ + . . . + P, . a" + . . . 



2 sin 



(21); 



(22). 



