§ 68. Rotation. Irregular Distribution. Irrotation, —Definition.—The propositions connecting boundary circulation and rotation include all cases of rotation, so that we know that however much the rotation differs in different parts of the fluid, the algebraic sum of the rotation taken over the whole of the region is equal to the integration of the circulation along the boundary (reckoned plus or minus, according to the law laid down).
Thus, the total rotation in a region containing fluid is zero when the sum of the circulation taken over a complete circuit of the boundary is zero; also, the motion of a fluid is "irrotational" when the sum of the circulation round a complete circuit of the boundary of its each small element is zero.
§ 69. Rotation. Mechanical Illustration.—In order to clearly dissociate the idea of rotation in a fluid from that of circular motion by virtue of which it may possess angular momentum, we may imagine a region of uniform rotation, such as that we have been considering, to have its motion intercepted by a net- work of rigid boundaries suddenly congealed throughout the region. Then the boundary system will at the instant of its formation receive an impulsive torque, and angular momentum of the rotating mass will be given up, but the rotation within the meshes of the network will persist, the new conditions being
those of the supposition in Fig. 37, the equal and opposite circulations along the boundaries in common being materialised. We can suppose a mechanical model constructed to represent this action. Let us imagine a frame mounted upon a shaft capable of revolution, and carrying a multitude of accurately balanced wheels mounted on frictionless bearings, these bearings being arranged parallel, and parallel to those of the main shaft. Let us suppose that the whole apparatus be initially rotating en bloc; then if we stop the motion of the frame each of the wheels will continue to spin with the same angular velocity as previously, and nothing that we can do with the frame will alter their rate of spin in the slightest. The frame corresponds
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