On the Vibrations of a Columnar Vortex. 155 



on again heating. For two tubes which were examined, this 

 time was in each case about 20 hours ; a shorter period 

 merely sufficed to diminish the depresion. The depression is 

 the result of an action between the liquid and the inner glass 

 surface of the capillary tube. 



3. Indications that surfaces exercise a slight action in deter- 

 mining the position at which the liquid condenses in the ex- 

 ternal tube have been observed. 



4. By reflecting a bright line of light from the apparently 

 convex and well-defined surface of ether in a tube of 20 millims. 

 diameter at a temperature near the critical, it may be inferred 

 to remain concave until it loses the power of reflecting when 

 it is plane. The apparent convexity is the result of refraction, 

 or, perhaps, of an action resembling mirage. 



5. The black ill-defined band which immediately succeeds 

 the disappearance of the liquid surface is the result of too 

 rapid heating, and possibly due to the mixing of liquid and 

 vapour when they are of nearly equal density. When very 

 slowly heated, as described, the defined concave surface is gra- 

 dually obliterated, and is last seen as a fine and often waving 

 line. Under this condition also the volume of the liquid at its 

 disappearance is greater than when it is rapidly heated. When 

 the liquid is vaporized by rapid heating, it has a higher tem- 

 perature and larger volume at the time of disappearance than 

 it has when first condensed by cooling: slowly heated and 

 cooled, these volumes and temperatures are more nearly the 

 same. 



Royal Indian Engineering College, 

 June 1880. 



XXIV. Vibrations of a Columnar Vortex. 

 By Sir William Thomson*. 



THIS is a case of fluid-motion, in which the stream-lines 

 are approximately circles, with their centres in one line 

 (the axis of the vortex) and the velocities approximately con- 

 stant, and approximately equal at equal distances from the 

 axis. As a preliminary to treating it, it is convenient to ex- 

 press the equations of motion of a homogeneous incompressible 

 inviscid fluid (the description of fluid to which the present in- 

 vestigation is confined) in terms of " columnar coordinates," 

 r, 6, z — that is, coordinates such that r cos 6 = x, rsin Q—y. 



If we call the density nnity, and if we denote by x, y, z the 

 velocity- components of the fluid particle which at time t is 



* From the Proceedings of the Royal Society of Edinburgh, March 1 

 1880. ' 



