Manchester Memoirs, Vol xlii. (1898), No. 9. 1 1 



where K^ is defined by 



-00 

 KJ^z) = / cos (2 sinh ti)du 



the latter series being of the "semi-convergent" class, 

 and therefore only available for large values of ^. For 

 small values of .s we may use the form 



K{^) =-{7+ log l^Voi^) + 5 + 2^2(1 - ^) + • ■ • (39), 



where 7 = -57721... . The condition analogous to (23), to 

 be satisfied at the external surface, is 



The effect of this is that, in our previous work, we must 

 replace 



by 



I,{va) _ K,[ra) 

 val^{va) vaKJ [vo) ' 



or, for sufficiently small values of va, 



—2 - log va, 



where only the most important term in each fraction has 

 been retained. When the wave-length is at all consider- 

 able as compared with the circumference of the tube, 

 the second fraction is small compared with the first ; ie., 

 the effect of the external fluid may be neglected. 



3. It has been assumed, so far, that the thickness of 

 the tube is small compared with the radius. When this 



• See Gray and Mathews. Bessel Fzinctions, pp. 67, 68. 



