1887.] 



Formation of Coreless Vortices in a Fluid. 



83 



ing, but time does not permit its being included in the present com- 

 munication. 



For Examples (5) and (6) the denominator of (10) is imaginary ; 

 and the proper modification, from (7) forwards, gives for these and 

 such cases, instead of (14), the following : — 



_ cos [>yOQ]+Bin jW0Q] (m 



The result is easily written down for each of the two last cases 

 i[ Examples (5) and (6)]. 



II. " On the Formation of Coreless Vortices by the Motion of a 

 Solid through an inviscid incompressible Fluid." By Sir 

 W. Thomson, Knt., LL.D., F.R.S. Received February 1, 

 1887. 



Take the simplest case : let the moving solid be a globe, and let 

 the fluid be of infinite extent in all directions. Let its pressure be of 

 any given value, P, at infinite distances from the globe, and let the 

 globe be kept moving with a given constant velocity, V. 



If the fluid keeps everywhere in contact with the globe, its velocity 

 relatively to the globe at the equator (which is the place of greatest 

 relative velocity) is fV. Hence, unless P>fV 2 ,* the fluid will not 

 remain in contact with the globe. 



Suppose, in the first place, P to have been >fV 3 , and to be 

 suddenly reduced to some constant value <|-V 2 . The fluid will be 

 thrown off the globe at a belt of a certain breadth, and a violently 

 disturbed motion will ensue. To describe it, it will be convenient to 

 speak of velocities and motions relative to the globe. The fluid must, 

 as indicated by the arrow-heads in fig. 1, flow partly backwards and 

 partly forwards, at the place, I, where it impinges on the globe, 

 after having shot off at a tangent at A. The back-flow along the 

 belt that had been bared must bring fco E some fluid ; and the free 

 surface of this fluid must collide with the surface of the fluid leaving 

 the globe at A. It might be supposed that the result of this collision 

 would be a "vortex sheet," which in virtue of its instability, would 

 get drawn out and mixed up indefinitely, and be carried away by the 

 fluid farther and farther from the globe. A definite amount of 

 kinetic energy would be practically annulled in a manner which I 

 hope to explain in an early communication to the Royal Society of 

 Edinburgh. 



But it is impossible, either in our ideal inviscid incompressible 

 * The density of the fluid is taken as unity. 



