BY SMALL OBSTACLES OF CYLINDRICAL AND SPHERICAL FORM. 261 



and if for all values of n > 0, 



[,_! (ha) + B B (n+ l)f n -i (to) = i"^"^., (ha) 



' ... (9). 



These equations (8) and (9) are sufficient to determine the various constants in the 

 expression (4) for the scattered sound. In the process of approximating to the values 

 of these constants we shall limit ourselves, as before, to the case when ha is a small 

 fraction ; in other words, we shall suppose that the radius of the obstacle is small 

 compared with the wave-length of the incident sound. We shall also suppose that 

 the gas in question is the air of the atmosphere. This will make crvfc 2 a small 

 fraction for all wave-lengths within the limits of audibility. 



With this assumption we may write, as before, 



/t = -(l-fu,o-/c 2 ) ......... (10). 



o 



Eliminating A n between the equations (9), we obtain 



B. {(n+ 1) 7Ay B+1 (ha)f^ (ka) +nf n - l (ha) AW/ B+1 (to)} 



= -*'A'.AW{^,. 1 (Aa)/ B+1 (Aa)-^ >+l (Aa)/.- 1 (*a)} . (11). 



Now with the help of a well known result in the theory of Bessel functions it may 

 be proved that 



*_! (ha)f n+l (Aa)-V/,, +1 (*)/._! (ha) = (2n+ 1) *-*+a-*. 

 Hence equation (11) takes the form 



(12). 



Retaining only the principal term in the coefficient of B n , we may write approxi- 

 mately 



whence we have 



to the same degree of approximation. 



Next eliminating B B between the equations (9), we obtain 



-D niH 2" . n ! 1 / 1 ~\ 



1 ' 



1 (ha)f^ (ka) + nf n . l (ha) Fa 2 / B+1 (ka)] 



-! (Aa) Fa 2 / B+1 (to) + (n + 1) A'aty,*, (Aa)/ B _ a (to)] . (14). 



