of Sound by Spheroids and Disks. 371 



v 2 and v 3 are usually required to the first order only, and 

 become 



%(«) = -4(1 — 3 cosh 2 a) . , ,. r f ro— To 



- v } o)^ J a smha(l — 3cosh J a) 2 



— T~~2 ( ^ cos ^ a + 1 — ^ cosh* a log coth b r ) 

 i» 3 (a)= ^— g-< 1 — 15cosh 2 a + 15cosh 3 a — 9 cosh a . log coth->- 



The calculation of these functions rapidly becomes laborious, 

 but enough have been found for our present purpose. 

 Their first approximations are proportional to the values of 

 Q n (cosha), where Q' m (/a) denotes the ordinary zonal harmonic 

 of the second kind. 



Asymptotic expansions of v n and w n . 



At a great distance from the obstacle, or at all points if it 

 be a sphere, v n and w n become Bessel functions. In fact, 

 under these conditions, HJ/a) may be regarded as P„(/a), and 

 therefore 



t \ t> / d \ sinO 

 * W=P fo) IT' 

 which has the value 



«.W ='"\/^ J » + i(^)> • • • • (27) 



Q 



where T = c cosh a, which is the major semi-axis of the con- 



focal through the distant point. 

 Thus w n admits the expansion 



/a n7r 



Vsin^-f). . ... (28) 



Now the Bessel function J_„_i(0), admitting an expansion 



satisfies J_=— -jl — ~ s = large positive integer, 



0+5)2 

 where the lower limit is the value selected for j3 in (24). 



