1 78 Lord Rayleigh on the Instability 



in which T represents the capillary tension,/) the density, and, 

 as usual, 



2 4 6 



Io(*)=JoO , *) = l+|j + 2^+ 2^12762 + - ••> ( 4 ) 



I 1 (^)=a / (^)=| + ^+^r^ + ... . (5) 



But if the fluid be outside the cylinder, Ave have to use that 

 solution of (1) for which the motion remains finite when r = oo . 

 This maybe expressed in two ways*. When r is great we 

 have the semi-convergent form 



9 \2krJ 6 L l.Skr + 1..8.(8*r)* 



1'.3».5' \ (n 



""TT2T3(8^) 3+ ---J ? ' • [b) 



and for all values of r the fully convergent series 



^. = ( 7 +logPr)I (^)-^S 1 -^,S 2 - (7) 



in which 7 is Euler's constant, equal to *5772 . « . , and 



Ml+S+J+-+i (») 



In this case the solution of the problem becomes 



' 7 "pa 3 <f>(ka) ...... l»J 



<£ being defined by (7). In (9) p represents the inertia of the 

 external fluid, that of the internal fluid being neglected, while 

 in the corresponding formula (3) p is the inertia of the internal 

 fluid, that of the external fluid being neglected. There would 

 be no difficulty in writing down the analytical solution ap- 

 plicable to the more general case where both densities are 

 regarded as finite. 



The accompanying Table gives the values of 



\{ & =&®y- m 



to which q in (3) is proportional, and of 



{^%F*y 0.) 



* See the writings of Sir G. Stokes j or ' Theory of Sound,' § 341. 



