246 MR. C. J. T. SEWELL: EXTINCTION OF SOUND IN A VISCOUS ATMOSPHERE 

 Consequently we obtain 



E n [ha . ka D/ (ha) D B ' (ka) - n* D n (ha) D n (ka}] = i" . (10). 



7T 



Now ha is a small quantity, and consequently we may use the approximations (7) ; 

 substituting for ~D n (ha) and D/(/J) from (7), we obtain 



_ 2!j k-n a -B n {ka D n ' (ka) + nV n (ka)} = ," 



or 



2*n 1 i _. T T\ n \ -o 4n 



-- . A "a " . ka D n _j (&a) B n = -- t". 



7T 77 



Hence we obtain as a first approximation 



O.n 1 J)"fi n 



B = _-fi_ __i _ _ 1 _ (11) 



" 2'- 1 ' (n-l) I' ka^.^ka) ' 



Similarly, by elimination of B n between equations (5), we obtain 



A n [Aa . ka D' n (/j) D', (Aa) -n 2 D n (ha) D n (fo)] 



= -2i" [/ta . AaJ', (Aa) D'. (Aa) -n 2 J n (Aa) D,, (ka)]. 



Using the approximate values given in (7), we obtain 



' D n+1 /.n 



-,^ 2 



or 



A _2i" 7r/t2 " CT2 '' D ^ (^) 

 2 2 "n!(n-l)!'D n _ 1 (A-a) 



for all integral values of n> 0. 



By a similar process,- but carrying the calculation to a higher degree of approxi- 

 mation by means of (8) and (9), we have 



Let us consider first the case when | ka | is small. In this case it will be sufficient 

 to derive the value of AI from (12). Hence we have approximately 



. 



DO (ka) 



Writing for convenience 



k = Xe- 1 '' 4 ", where X = (cr/p) 1 / 2 ....... (14), 



