Vibrations of a Columnar Vortex. 157 



where p, t, iv, and ts are functions of r, each infinitely small 

 in comparison with T. Substituting in (4) and (5) and neg- 

 lecting squares and products of the infinitely small quantities, 

 we find 



- c ^ = (n-i-) -2-T 

 dr \ r/P r , 



iv / T\ /T dT\ 



+ m / ur 



dp o it A 



dr r r 



(7) 



(8) 



Taking (7), eliminating -a, and resolving for p, r, we find 



1 / .T\f/ .T\dw »/T,dT\ \ 1 



1 f/T dTV .T\Ap ,irT 2 <ZT 2 / .TVl 1 L 



where D= 8T/T rfT\_ / _.T\« 



r V? 1 ar / \ y / 



For the particular case of ra = 0, or motion in two dimen- 

 sions (r, #), it is convenient to put 



^=£ (10) 



In this case the motion which superimposed on r= and r# = T 

 gives the disturbed motion is irrotational, and </>sin (nt — id) 

 is its velocity-potential. It is also to be remarked that, when 

 m does not vanish, the superimposed motion is irrotational 

 where, if at all, and only where T= const. /r; and that when- 

 ever it is irrotational, (/>, as given by (10), is its velocity- 

 potential. 



Eliminating p and r from (8) by (9), we have a linear dif- 

 ferential equation of the second order for iv. The integration 

 of this, and substitution of the result in (9), give iv, p, and t 

 in terms of r, and the two arbitrary constants of integration 

 which, with m, n, and ?', are to be determined to fulfil what- 

 ever surface-conditions, or initial conditions, or conditions of 

 maintenance are prescribed for any particular problem. 



Crowds of exceedingly interesting cases present themselves. 

 Taking one of the simplest to begin: — 



Case I. 

 Let T=cor (w const,), (11) 



