Vibrations of a Columnar Vortex. 161 



JT 2 dr 



np2 



P-i.(r-«) 



= 3 — — r-^ cos w&s cos (n£ — ?' 0) . • (32 ) 



Hence, and by (6), and (26), and (25), and (23), the condition 

 P +p = at the free boundary gives 



i [OF, (ma) + €¥i (ma)] + fo=gff [CL(ma) + Cfc(ma)] = 0. 

 a m .... (33) 



Eliminating C/C from this by (27), we get an equation to 

 determine n, by which we find 



wr-^+vN), (34) 



where N is an essentially positive numeric. 



II. — Subcase. 

 A very interesting Subcase is that of a = co , which, by 

 (27), makes 0=0, and therefore, by (33), gives 



X=*»^Pg (35) 



t(ma) 



Whether in Case II. or Subcase II., we see that the dis- 

 turbance consists of an undulation travelling round the cylin- 

 der with angular velocity 



(l+ ^)„. (l -^), 



or of two such undulations superimposed on one another, tra- 

 velling round the cylinder with angular velocities greater than 

 and (algebraically) less than the angular velocity of the mass 

 of the liquid at its free surfaces by equal differences. The 

 propagation of the wave of greater velocity is in the same di- 

 rection as that in which the liquid revolves ; the propagation 

 of the other is in the contrary direction when N > i 2 (as it 

 certainly is in some cases). 



If the free surface be started in motion with one or other of 

 the two principal angular velocities (34), or linear velocities 



ao> 1 1 ± — — ), and the liquid be then left to itself, it will per- 

 form the simple harmonic undulatory movement represented 

 by (6), (26), (23). But if the free surface be displaced to the 

 corrugated form (30), and then left free either at rest or with 

 Phil. Mag. S. 5. Vol. 10, No. 61. Sept. 1880, N 



