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

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.^[1] 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

1. The velocity potential may fall to zero in a portion of the fluid in the course of its motion without that portion of the fluid losing the attribute of velocity potential in the sense of Lagrange's theorem.

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