Then if r be the radius of the whole enclosure a a, and n r be the radius of the region e e, and x the total rotation, the rotation within the region e e as measured by the circulation along its boundary will be n2 χ, the remaining rotation in the region between the boundaries will therefore be χ —n2χ that is to say, the circulation along the external surface of the boundary e e is equal and opposite in sign to that along its internal surface.
Now if we regard the rotation of the fluid mass as a matter of rigid dynamics, the motion in the path e e is the same in sense whether it takes place in the matter external or internal, and in general rotation is an algebraic quantity, measured plus or minus, according to whether it takes place counterclockwise or clockwise (the latter being taken minus by convention). It is evident, however, that circulation along a boundary (also an algebraic quantity) cannot be so measured, but is plus or minus according as the fluid flows towards the right or the left hand of an observer stationed on the boundary facing the fluid. Thus, in the simple case illustrated, let us suppose the rotation to be positive (counterclockwise), then to an observer stationed on the "mainland" the circulation will pass from left to right, and is reckoned positive. If the observer now take his stand on the "reef" e e, and face the outer basin, the circulation will pass from right to left, and is therefore negative. If he now turn about and face the inner basin, the circulation is from left to right, and is positive. Another method of defining the sense of a circulation is to suppose an observer swimming in the fluid to keep the boundary always on his right hand, then the direction in which he is swimming is positive and the opposite direction negative. The positive direction is indicated by arrows in the figure.
Rotation in a fluid as above defined is a conception apart from any quantity known in rigid dynamics, and owes its importance to certain propositions relating to fluid motion. It is a quantity that in a perfect fluid can undergo no change. Conservation of rotation is an absolute law in an inviscid fluid.
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