110 LORD RAYLEIGH ON THE ACOUSTIC SHADOW OF A SPHERE. 



where 



i 2 sin 20 /<>> 



3 2 = 1 - 2 2 cos 20 + t\ tan = -^ rZa - (<>) 



For our purpose it is only necessary to write C/t and S/< for C and S respectively, 

 and to identify t~ with x in (y). Thus 



sn 



(e), 



and <f> being given by (3). We find, with t 1 ^ 



1 



I 



so that 



and 



whence 



3 2 = 4sin 2 0[l - 



= 2 sin 



1 

 8 



(0; 



sin 20 

 tan ^ == 2si,r0 + l/2 



cot < 



4?;. 



Using (), 



in (e) we get 



^ Jl-ll.oosj 



n sin 0)1 4J 



TT COt 



, - Q 



on 



}... 



which is the expression required. 



By this extension, not only is a closer approximation obtained, but the logic of the 

 process is improved. 



A comparison of values according to (6) with the true values may be given in the 

 case of n equal to 20. 



VALUES of P 20 . 







