RECENT PROGRESS IN HYDRODYNAMICS. 67 



angle between q and w, and v is the normal distance at the point of the 

 two surfaces. 



The problem of the most general vibration of a straight columnar 

 vortex of constant vorticity has been treated by Thomson.' If fluid be 

 revolving irrotationally in a fixed cylinder, there -will be a cylindric space 

 along the axis, either empty, or filled with fluid in rotational motion. In 

 the former case, suppose the surface disturbed so that the generating 

 lines and normal sections are deformed into harmonic curves of difFei'ent 

 Avave-lengths, the wave-lengths of the cross section being of course sub- 

 multiples of the undisturbed circumference. Then Thomson shows 

 that two velocities of propagation are possible, of the form — angular 

 velocity = iw {i i: s/Nyji where iv is the angular velocity of points on the 

 inside surface of the fluid, i is the number of crests in a normal section, 

 and N depends only on the number of crests in unit length along a 

 generating line. One set, therefore, travels in the same direction as the 

 rotation, the other in the same or opposite direction, according to the 

 magnitude of N. For the special case, where the containing vessel is 

 infinitely large, and the distance between the crests on a generating line 

 large compared with the circumference of the hollow N=^l -}- Jc- ('IISO 

 — log Ic) where h denotes this ratio. Here the slow wave for the case 

 i = l travels in the reverse direction to the rotation. Of more importance 

 than the foregoing is the case of the small vibrations of a vortex column 

 in an infinite irrotationally moving liquid. The periodic times for given 

 harmonic initial disturbance are given by the roots of a transcendental 

 equation, which has an infinite number of roots. This is much simplified 

 when. the disturbance is only longitudinal. When the distance between 

 successive swellings of the core is large compared with the circumference, 

 the velocity of propagation corresponding to the smallest root is about f 

 of the circular velocity at the surface of the vortex. When there is no 

 longitudinal displacement, the angular velocity of sectional waves is (i — 1) 

 X ang. vel. of the vortex, where i denotes as before the number of crests 

 on the circumference. The time of pulsation of a hollow vortex enclosed 

 in a flexible cylindrical shell, over whose surface the pressure is constant 

 has been given by the writer. - 



Mr. F. D. Thomson ^ has proved an interesting theorem relating to a 

 steady motion within cylindrical surfaces. Suppose a cylindrical vessel, 

 of any sectional form, and the contained fluid to be rotating as a rigid body, 

 and that the vessel itself is suddenly brought to rest, then the resulting 

 motion of the fluid will be steady. Another proof has been given by 

 Stokes,* but it is easily seen to be a consequence of the uniform distribution 

 of vorticity and Helmholtz' laws. 



Quite lately, two papers have been published respectively by Craig ^ 

 and Rowland^ almost simultaneously. They both seem to have been 

 struck by the fact that if the same operations are performed on the com- 

 ponents of rotation that are performed on the components of velocity to 

 deduce the rotations, functions are obtained which satisfy the equation of 



' 'Vibrations of a columnar vortex,' PJdl. Muff. (.5), x. p. 15!). 

 ^ Proc. Camb. Phil. Soc, iii. p. 28.S. 



= ' Some cases of fluid motion,' Ox. Cam. Diih. Mess. Math. iii. p. 238, iv. p. 37. 

 ♦ Reprint, vol. i. p. 7 (1880). 



' ' Methods and results, Sec' United States Coast Survey, Washington, 1881. 

 ^ 'On the motion of a perfect incompressible fluid when no solid bodies are 

 present,' Amer. Jour, of Math, iii, p. 226. 



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