12 



PROFESSOR ANDREW GRAY, 



If, further, the surface be so far from the vortices that the fluid is at rest [or, wh 

 this is not the case, the motion be steady], we know that 



/¥*=»• 



Hence in either of these cases X A = X B . We have then 



I di dx ~ I ( 2r £~ 2wr >) d * =o 



en 



and we see that 



AB 



AB 



I jf^ dxd y dz - j f f( 2v £ - lio-qjdxdydz = 



(29); 



(30), 



where the volume integrals are taken throughout the fluid. In the case of steady 

 motion this ffives 



Similarly, 



I f f (2v£ - 2wy } )dxdydz = 0" 

 j f f(2w£ - 2uQdxdydz = 

 f f f(2u v -2v£)dxdydz = 



(31). 



These equations are usually obtained only for the case of the fluid contained 

 within a surface which is at every point infinitely distant from the vortex system. 

 D Unless there is some error in the reasoning here stated this 

 restriction would appear to be unnecessary. 



20. A vortex-line is defined as a curve the tangent to which 

 at any point is in the direction of the resultant rotation at 

 the point. Since d-^/ds' is zero when ds' is taken along a stream- 

 line or along a vortex-line, it is zero for every direction in the 

 plane determined at any point by the stream-line and the 

 vortex-line through the point. It has therefore a maximum 

 value at the point along a normal to the plane contained by 

 these directions. 



A tubular space bounded by vortex-lines and filled with 

 fluid in elemental rotation is called a vortex-tube. Consider 

 a portion of a vortex-tube between two cross- sections formed by 

 ds, an element of a stream- line, and ds f , taken at right angles 

 to the plane of the stream-line and a vortex-line at a point 

 on the surface of the tube. Such a portion of a tube is shown in the diagram by 

 P'A' CD' BQ. P'A' and QB are taken at right angles to the planes of the stream- 

 line and vortex-line at P' and Q, and represent ds' taken at these points. P'Q, 

 A'B, CD' are vortex-lines, and P'C, QD' represent ds taken at the same points. 

 Since ds is not necessarily at right angles to the vortex-line at P' or Q, let PC, QD 

 be so taken. Then in PAC DBQ we have a portion of a vortex-tube with its 



