Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/112

§ 70 of the fluid the pressure difference applied between the points $$a$$ and $$b$$ requires to be the same as that between $$c$$ and $$d$$, so that the normals $$a\ c$$ and $$b\ d$$ to the field of flow are equipotentials ($$\phi =$$ constant).

This demonstration may be taken as applied to every small element of the field, so that the proposition is proved.

Corollary: When a fluid has velocity potential its motion is irrotational.

§ 71. Physical Interpretation of Lagrange's $$\boldsymbol{\phi}$$ Proposition.—The foregoing proposition, taken in conjunction with that relating to the conservation of rotation, constitutes a demonstration of Lagrange's theorem that “If a velocity potential exist at any one instant for any finite portion of a perfect fluid in motion under the action of forces which have a potential, then, provided the density of the fluid he either constant or a function of the pressure only, a velocity potential exists for the same portion of the fluid at all instants before or after.”

This statement, save to a mathematician, is not very clear, as it is difficult to obtain a sufficiently close conception of velocity potential to be able to attach any physical meaning to its conservation. The inversion of the statement, however, obviates all difficulty; it then becomes: If the motion of any portion of a perfect fluid he irrotational at any instant of time, then, provided the density of the fluid he either constant or a function of the pressure only, the motion of the same portion of the fluid will he irrotational at all instants before and after.

§ 72. A Case of Vortex Motion.—The case of cyclic motion resulting from an interchange of the functions $$\psi$$ and $$\phi$$ in the source or sink system is one of particular interest. If (Fig. 35) we suppose the origin circumscribed by a line of flow, then we