in which p is the density of the fluid, assumed uniform, 11 is the particle velocity, and p^ the 

 pressure in the fluid at infinity. In many figures the difference, V-p^ or p (C/-^ -g^yo, along 

 selected lines or curves is shown on an arbitrary scale. The pressure along the y-axis is some- 

 times plotted horizontally, with positive values toward the right. Any case in which the bound- 

 aries are in uniform translatory motion may be reduced to the corresponding case in which they 

 are at rest by a suitable change of the frame of reference, or by imparting to everything an equal 

 velocity in the opposite direction. Such a cliange does not alter the distribution of pressure or 

 the forces. 



Results will commonly be stated in terms of a particular choice of axes, and sometimes 

 in terms of particular units of length and of velocity. The use of special units permits the math- 

 ematical developments to be made in compact form; but the equations may not be dimensionally 

 balanced. It is to be understood that the formulas, if too specialized, may always be general- 

 ized by substituting A 2, k.x, k^y, k.T for s, x, y, r, also k^w, k„(f), k„il/, for w, <f), lA, and k^u/k^, 

 k^v/k^, k„q/k^, for u,v,q, where k^ and k^ are any real numbers, provided these changes 

 are made consistently in all formulas. Velocities are thereby changed in the ratio k^/k^, since 

 u, V, q are then given by the original expressions each multiplied by k^/k^\ and all linear 

 dimensions similarly become 1/k^ times as great. Even the velocity at infinity is changed in 

 the ratio k./k ; and the kinetic energy in a layer of unit thickness perpendicular to the planes 

 of flow or T. is multiplied in proportion to velocity squared times area or by l/k^. The dimen- 

 sional balance may then be restored, if desired, by assigning the proper dimensions to k^ and 

 k^. One type of change without the other may be made by letting either k^ or k^ be unity. 



In addition, of course, tiie axes may be moved into any other position by means of the 

 usual formulas. The method of doing this in terms of 2 is important and was explained in 

 Section 25. To displace the flow and all boundaries through distances h^ in the ^-direction 

 and h in the y-direction, without rotation, it suffices to replace x by x-h^ and y by y-h^ in 

 all formulas, or 2 by 2 -A where h = h^ +^^2* ^° rotate everything through an angle a about the 

 origin, which requires rotation of the axes in the opposite direction relatively to the flow field, 

 replace a; by a; cos " + y sin a and y oy - x sin " + y cos a or 2 by ze~ "^, and u and v, there- 

 fore, by u cos a + v sin a and - u sin a + v cos a , respectively, in all formulas. To effect 

 first the rotation, about the initial origin, then the displacement, substitute {z-h)e~'^ for 2; 

 or, if 2 = f(w), take z = h + e^" fi'i^)' 



Where 10(2) contains a real multiplicative constant, often A or U, it is to be understood 

 that reversal of the sign of this factor merely reverses all velocities, with an accompanying 

 change of sign of (f> and lAbut witliout any change in the geometrical equipotential curves and 

 streamlines and without change of the pressure. Arrows drawn on tlie streamlines in the plots 

 refer in each case to a positive value of this constant. 



It should be remembered that states of flow of an incompressible fluid may be superposed 

 freely to form new states of flow. The potential, stream functions, and velocity components 

 add algebraically, the velocities themselves, vectorially. The pressures and forces, however, 



73 



