Vibrations of a Columnar Vortex. 159 



The problem thus solved is the finding of the periodic disturb- 

 ance in the motion of rotating liquid in a space between two 

 boundaries which are concentric circular cylindric when un- 

 disturbed, produced by infinitely small simple harmonic normal 

 motion of these boundaries, distributed over them according 

 to the simple harmonic law in respect to the coordinates z, 0. 

 The most interesting Subcase is had by supposing the inner 

 boundary evanescent (a = 0), and the liquid continuous and 

 undisturbed throughout the space contained by the outer cy- 

 lindric boundary of radius a. This, as is easily seen, makes 

 iv = Q when r = 0, except for the case i=l, and essentially, 

 without exception, requires that c be zero. Thus the solution 

 for w becomes 



w=GJi(yr), (19) 



or 



ic = VIi(ar)i (20) 



and the condition p = c when r — a gives, by (13), 



o= *l , . . . (21) 



or the corresponding I formula. 



By summation after the manner of Fourier, we find the 

 solution for any arbitrary distribution of the generative dis- 

 turbance over the cylindric surface (or over each of the two if 

 we do not confine ourselves to the Subcase), and for any arbi- 

 trary periodic function of the time. It is to be remarked that 

 (6) represents an undulation travelling round the cylinder 

 with linear velocity na/i at the surface, or angular velocity 

 n/i throughout. To find the interior effect of a standing vi- 

 bration produced at the surface, we must add to the solution 

 (6), or any sum of solutions of the same type, a solution, or a 

 sum of solutions, in all respects the same, except with — n in 

 place of ft. 



It is also to be remarked that great enough values of i make 

 v 2 negative, and therefore v imaginary ; and for such the solu- 

 tions in terms of cr and the I,-, fc functions must be used. 



Case II. — Hollow Ir rotational Vortex in a fixed Cylindric 



Tube. 

 Conditions : — 



T=^,-.=0when,.= «; ] 



and -p+p = for the disturbed orbit, r=n+jr a dt,J 



