250 SIR WILLIAM THOMSON ON VORTEX MOTION. • 



But a, /3, y are the projections of A on the planes of the pairs of the rectangular 

 axes ; and so the proposition is proved. 



It follows, of course, that the composition of rotations in a fluid fulfils the 

 law of the compositions of angular velocities of a solid, of linear velocities, of 

 forces, &c. 



CO. (g). Hence, in any infinitesimal part of the fluid, the circulation is zero in 

 the periphery of every plane area passing through a certain line ; — the resultant 

 axis of rotation of that part of the fluid. But (a) the circulation remains zero in 

 every closed line moving with the fluid, for which it is zero at any time. Hence 



(h). The axial lines [defined (i)] move with the fluid. 



(i). Definition. An axial line through a fluid moving rotationally, is a line 

 (straight or curved) whose direction at every point coincides with the resultant 

 axis of rotation through that point. 



(/). Proposition. The resultant rotation of any part of the fluid varies in 

 simple proportion to the length of an infinitesimal arc of the axial line through 

 it, terminated by points moving with the fluid. To prove this, consider any in- 

 finitesimal plane area, A, moving with the fluid. Let w be the resultant rotation, 

 and 6 the angle between its axis and the perpendicular to the plane of A. This 

 makes w cos d the component rotation in the plane of A ; and therefore Aw cos 6 

 remains constant. Now, draw axial lines through all points of the boundary of 

 A, forming a tube whose area of normal section is A cos 6. The resultant rota- 

 tion must vary inversely as this area, and therefore (in consequence of the in- 

 compressibility of the fluid) directly as the length of an infinitesimal line along 

 the axis. 



(k). Form a surface by axial lines drawn through all points of any curve in 

 the fluid. The circulation is zero round the boundary of any infinitesimal area 

 of this surface ; and therefore (b) it is zero round the boundary of any finite 

 area of it. 



(I). Let the curve of (k) be closed, and therefore the surface tubular. On this 

 surface let ABCA, A'B'C'A' be any two curves closed round the tube, and ADA' 

 any arc from A to A'. The circulation in the closed path, ADA'B'C'A'DACBA, 

 is zero by (h). Hence the circulation in ABCA is equal to the circulation in 

 A'B'C'A' — that is to say, 



The circulations are equal in all circuits of a vortex tube. 



(?n). Definitions. An axial surface is a surface made up of axial lines. A 

 vortex tube is an axial surface through every point of which a finite endless path, 

 cutting every axial line it meets, can be drawn. Any such path, passing just 

 once round, is called a circuit, or the circuit of the tube. The rotation of a vortex 

 tube is the circulation in its circuit. A vortex sheet is (a portion as it were of a 

 collapsed vortex tube) a surface on the two sides of which the fluid moves with 

 different tangential component velocities. 



