23. As a rule these equations are established immediately by determining the component acceleration of the fluid particle which is passing through (x, y, z) at the instant t of time considered, and saying that the reversed acceleration or kinetic reaction, combined with the impressed force per unit of mass and pressure-gradient, will according to d'Alembert's principle form a system in equilibrium. To determine the component acceleration of a particle, suppose F to denote any function of x, y, z, t, and investigate the time rate of F for a moving particle; denoting the change by DF/dt, DF F(x + u[delta]t, y + v[delta]t, z + w[delta]t, t + [delta]t) - F(x, y, z, t) -- = lt·---------------------------------------------------------------------------- dt [delta]t dF dF dF dF = -- + u-- + v-- + w--; (1) dt dx dy dz and D/dt is called particle differentiation, because it follows the rate of change of a particle as it leaves the point x, y, z; but dF/dt, dF/dx, dF/dy, dF/dz (2) represent the rate of change of F at the time t, at the point, x, y, z, fixed in space. The components of acceleration of a particle of fluid are consequently Du du du du du -- = -- + u-- + v-- + w--, (3) dt dt dx dy dz Dv dv dv dv dv -- = -- + u-- + v-- + w--, (4) dt dt dx dy dz Dw dw dw dw dw -- = -- + u-- + v-- + w--, (5) dt dt dx dy dz leading to the equations of motion above. If F (x, y, z, t) = 0 represents the equation of a surface containing always the same particles of fluid, DF dF dF dF dF -- = 0, or -- + u-- + v-- + w-- = 0, (6) dt dt dx dy dz which is called the differential equation of the _bounding surface_. A bounding surface is such that there is no flow of fluid across it, as expressed by equation (6). The surface always contains the same fluid inside it, and condition (6) is satisfied over the complete surface, as well as any part of it. But turbulence in the motion will vitiate the principle that a bounding surface will always consist of the same fluid particles, as we see on the surface of turbulent water. 24. To integrate the equations of motion, suppose the impressed force is due to a potential V, such that the force in any direction is the rate of diminution of V, or its downward gradient; and then X = -dV/dx, Y = -dV/dy, Z = -dV/dz; (1) and putting dw dv du dw dv du -- - -- = 2[xi], -- - -- = 2[eta], -- - -- = 2[zeta], (2) dy dz dz dx dx dy d[xi] d[eta] d[zeta] ----- + ------ + ------- = 0, (3) dx dy dz the equations of motion may be written