Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/115

Rh intersections athwart the stream will be the sum of the components.

§ 74. The Method of Superposed Systems of Flow.—The conception on which the foregoing method has been based can only be applied so long as the fluid moves en masse, but it can be shown that the method is applicable to all cases of irrotational motion. If we confine our attention to the field of force developed at the instant of application of the component impulses, then it is clear that the resultant field can be obtained by the use of the parallelogram of forces as shown in the figure; there is, however, another, and perhaps more convincing, method of proof; this is the method of superposition.

Let us suppose that instead of two motions being superposed on one fluid current two fluid currents be superposed on one another. This is at first difficult, owing to the instinctive but wholly imaginary difficulty of regarding it as possible for two bodies to occupy the same space at the same time. To simplify ideas, let us suppose the motion to be two-dimensional, so that it may be fully represented on a plane surface; then if we represent one motion on one plane and another motion on a plane adjacent to it the two systems will be superposed; and further, if we take as many systems as we wish and represent them on as many adjacent planes they become superposed. And since a plane possesses no thickness, such superposed systems, however numerous, occupy no finite quantity of the third dimension, and in fact constitute but one plane.

Now, reverting to the argument, let us suppose that any two systems of fluid motion be superposed one on the other. Then so long as we can identify the particles belonging to each separate system (as supposing the streams to consist of different kinds of matter), the two systems must be regarded as separate; but if we imagine that we cannot distinguish the matter in the one stream from that in the other, then a flux across any imaginary barrier in one direction will neutralise an equal flux across the same