Let us take the case of a cylindrical body of fluid rotating en masse about its axis. Then we may regard such motion as approximately composed of a number of cyclic motions superposed, and with their internal boundaries removed. Let us assume the cylindrical space to be subdivided by a number of concentric cylindrical surfaces, such as the lines of flow of a cyclic system, and, beginning at the centre, let us suppose a cyclic system to be started about a filament so that the velocity at the surface of the filament is that of the rotation. Then, taking the next concentric surface and treating it as a boundary, let us suppose a further cyclic system to be superposed on the first so that the velocity at the surface in question becomes that of the rotation, and again with the next concentric surface, and so on; then by taking the concentric surfaces sufficiently close to one another the motion of the fluid in rotation can be approximated to any desired degree. So long as the boundaries be supposed to exist the system is a superposed series of cyclic motion; if the boundaries be supposed withdrawn the motion is one of uniform rotation.
Now let us suppose such a system superposed on a motion of translation. Each cyclic system will give rise to a transverse resultant force on its boundary so that we shall have forces acting throughout the fluid occupied by the rotation. It is here assumed that the fluid is constrained to follow the paths of motion as geometrically laid down as the result of superposition, and it is shown that such constraint involves forces acting from without distributed over the whole region occupied by the rotation, a thing which under the conditions of the hypothesis is impossible of achievement.
The impossibility of compounding rotational motion with translation otherwise follows directly from Lagrange's theorem, for the resultant would involve the transfer of rotation from one part of a fluid to another, and would thus involve the violation of a principle that is fundamental.
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