94 



LORD EAYLEIGH ON THE ACOUSTIC SHADOW OF A SPHERE, 



In the case of = 0, or \j, = + 1, the P's are all equal to + 1, and we have 

 nothing more to do than to add together all the terms in the above table. When 

 Q = 180, or p. = 1, the even P's assume (as before) the value + 1, but now the 

 odd P's have a reversed sign and are equal to 1. If we add together separately 

 the even and odd terms, and so obtain the two partial sums S : and S 2 , then 2^ + S 3 

 will be the value of S for = 0, and 2j S 3 will be the value of 2 for = 180. 

 And this simplification applies not merely to the special values and 180, but to all 

 intermediate pairs of angles. If Sj -f- S 3 corresponds to 0, Sj S 2 will correspond to 

 180 - 6. 



For and 180 we find 



2 t = + 1-22870 + -35326 i, 



S 3 = + 0-31135 -f -85436 i; 

 whence for = 



S = 2 (F + iG) = + 1-54005 + 1 '20762 i, 

 and for = 180 



S = 2(F + iG) = + 0-91735 - 0-50110?. 



When = 00, the odd P's vanish, and the even ones have the values 



. . i 



" 



1.3 



2. 4 



P = -2:4T6> fe - 



For other values of we require tables of P n (0) up to about n = 20. That given 

 by Professor PERRY* is limited to n less than 7, and the results are expressed only to 

 4 places of decimals. I have been fortunate enough to interest Professor A. LODGE 

 in this subject, and the Appendix to this paper gives a table calculated by him 

 containing the P's up to n = 20 inclusive, and for angles from to 90 n at intervals 

 of 5. As has already been suggested, the range from to 90 practically covers 

 that from 90 to 180, inasmuch as 



P 2/ , (90 + 0)= P 2S (90 - 0), P, u+l (90 + 0) = - P 2 ,, +I (90 - 0). 



* 'Phil. Mag.,' vol. 32, p. 516, 1891 :. see also FARR, vol. 49, p. 572, 1900. 



