The Theory of Sound, Vol. II

impressed forces, and the reactions against acceleration. In the limit, when the tetrahedron is taken indefinitely small, the fluid pressures on its sides become paramount, and equilibrium requires that their whole magnitudes be proportional to the areas of the faces over which they act. The pressure at the point x, y, z will be denoted by p.

237. If ρXdV, ρYdV, ρZdV, denote the impressed forces acting on the element of mass ρdV, the equation of equilibrium is

where dp* denotes the variation of pressure corresponding to changes dx*, dy*, dz* in the coordinates of the point at which the pressure is estimated. This equation is readily established by considering the equilibrium of a small cylinder with flat ends, the projections of whose axis on those of coordinates are respectively dx*, dy*, dz*. To obtain the equations of motion we have, in accordance with D'Alembert's Principle, merely to replace X, etc. by X − Du*/Dt*, etc., where Du*/Dt*, etc. denote the accelerations of the particle of fluid considered. Thus

dp*/dx* = ρ (X - Du*/Dt*)
dp*/dy* = ρ (Y - Dv*/Dt*) . . . . . . . . . . . . . . . . . . . (1).
dp*/dz* = ρ (Z - Dw*/Dt*)

In hydrodynamical investigations it is usual to express the velocities of the fluid u, v, w in terms of x, y, z and t. They then denote the velocities of the particle, whichever it may be, that at the time t is found at the point x, y, z. After a small interval of time dt*, a new particle has reached x, y, z; du*/dt* · dt* expresses the excess of its velocity over that of the first particle, while Du*/Dt* · dt* on the other hand expresses the change in the velocity of the original particle in the same time, or the change of velocity at a point, which is not fixed in space, but moves with the fluid. To this notation we shall adhere. In the change contemplated in d/dt*, the position in space (determined by the values of x, y, z) is retained invariable, while in D/Dt* it is a certain particle of the