The equality of 2ΣωdS for the surface to the circulation round its edge was expressed by Thomson as an analytical theorem of integration, which was first given by Stokes in a Smith's Prize paper set in 1854. It is here stated, apparently by an oversight, that it was first given in Thomson and Tait's _Natural Philosophy_, § 190. In the second edition of the _Natural Philosophy_ the theorem is attributed to Stokes. It is now well known as Stokes's theorem connecting a certain surface integral with a line integral, and has many applications both in physics and in geometry. Now consider the resultant angular velocity at any point of the fluid, and draw a short line through that point in the direction of the axis of rotation. That line may be continued from point to point, and will coincide at every one of its points with the direction of the axis of rotation there. Such an axial curve, as it may be called, it is clear moves with the fluid. For take any infinitesimal area containing an element of the line; the circulation round the edge of this area is zero, since there is no rotation about a line perpendicular to the area. Hence the circulation along the axial curve is zero, and the axial curves move with the fluid. Take now any small plane area dS moving with the fluid, and draw axial lines through every point of its boundary. These will form an axial tube enclosing dS. If θ be the angle between the direction of resultant rotation and a perpendicular to dS, the cross-section of the tube at right angles to the normal, and to the axial lines which bound it, is dS.cosθ. Let these axial lines be continued in both directions from the element dS. They will enclose a tube of varying normal cross-section; but the product of rotation and area of normal cross-section has everywhere the same value. A vortex-tube with the fluid within it is called a vortex-filament.