Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/114

motion is compounded of two component motions whose velocity and direction are known. Then it is evident that the two component motions can be compounded by drawing a parallelogram, which may either be regarded as a “parallelogram of velocities” if we take its elements to represent velocity, or a “parallelogram of forces” if we take its elements to represent the impulses by which the motion is produced. Thus, if we compound a north wind with an east wind having the same velocity, the result is a north-east wind having a velocity ${\sqrt {2}}$ times as great; and the forces that would produce the two air currents separately would produce the combined current if acting simultaneously.

Fig. 39. If we denote the strength of each superposed stream by a series of parallel lines, so that the flux or quantity of fluid passed per unit time is the same at every point between each adjacent line and its neighbour—that is to say, if we draw the lines of flow, $\psi =$ constant, for each component stream, then the distance separating any two adjacent lines will be inversely as the velocity,^[1] and the network formed by the superposed systems will give the parallelogram of velocities at every point. This method of compounding the two systems of flow is illustrated in Fig. 39, in which $a\ a\ a$ and $b\ b\ b$ represent the component streams, and $c\ c\ c$, drawn diagonally, gives the resultant flow. It is evident that the lines $c\ c\ c$ will quantitatively represent equal values of $\psi$, for the resultant flux across any line $d\ d$ drawn through

1. Referring to the diagram to the right of Fig. 39, we have from geometrical considerations the normals g and j respectively proportional to the sides of the parallelogram h and k.

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