102 LORD RAYLEIGH ON THE ACOUSTIC SHADOW OF A SPHERE, 



EXPLANATION OF THE METHODS OF COMPILING AND CHECKING THE ABOVE 



TABLES. 



Calculation of the Even Orders. 



The Zonal Harmonics of even order in the foregoing tables were calculated from 

 the formula obtained by expanding (1 2x cos 6 + x")'* by means of the form 



( + ape* + ...+ <*&#* + ...+ o^V" + . . .) 

 where 



i q F. /o _. i \ 



J.*Ot/**ljU/JLf -i 



2 . 4 . 6 . . . 2r 



whence 



P 2j! ($) = a,~ + 2a n _ l a n + l cos 20 + . . . + 2 a 2n cos 20. 



To calculate the coefficients a,,", 2a,,_ ] r/, / , + 1 , . . . 2a,,r 2 , 1 , an auxiliary table of values 

 of log ]0 , was formed from r = 1 to ? = 20, to 8 decimal places ; and a similar table 

 of log ]0 2, from r = to ? = 9 ; so as readily to combine them to form (to 7 decimal 

 places) the logarithms of the required coefficients for different values of r>. 



The coefficients were then calculated to 7 decimal places from their logarithms, and 

 checked for each value of n by seeing that they added up to unity in each case. 



Next, a table of values of log cos 26, log cos 46, ... to 7 decimals, was formed for 

 all values of 6, at 5 intervals, from 5 to 90. The addition of these to the 

 logarithms of the corresponding coefficients gave the logarithms of the various terms 

 (except as regards sign) in the above expansion of P 2ll (ff). From these logarithms 

 the terms themselves were calculated to 7 decimals and tabulated, the positive terms 

 in black, and the negative terms in red ink. The accuracy of these terms was 

 checked by making use of the identities 



(1.) 2 cos 60 = 1, 

 (2.) cos 50 + cos 70 = cos 10, 

 (3.) cos 40 + cos 80 = cos 20. 

 This, in addition to the primary identity 



checked all the terms effectually except those which were multiples of cos 30., 

 These were checked by adding a number of them together and comparing their sum 

 with the sum of the coefficients multiplied in a lump by cos 30. 



