three equations (26) Art. 18 in order by da, db, dc, and adding, or, with our present notation, udx + vdy+wdz = d (p. + x)= dp, say; which proves the theorem. 0 It is to be particularly noticed that this continued existence of a velocity-potential is predicated, not of regions of space, but of portions of matter. A portion of matter for which a velocitypotential exists moves about and carries this property with it, but the portion of space which it originally occupied may, in the course of the motion, come to be occupied by matter which did not originally possess this property, and which therefore cannot have acquired it. The above theorem, stated in an imperfect form by Lagrange in Section XI. of the Mécanique Analytique, was first placed in its proper light by Cauchy. Other proofs, to be reproduced further on, have since been given by Stokes *, Helmholtz, and Thomson. A careful criticism of Lagrange's and other proofs has been given by Stokes *. 24. The class of cases in which a velocity-potential exists includes all those where the motion has originated from rest under the action of forces of the kind here supposed; for then we have, initially, or udx+vdy+wdz = 0, $ = const. Again, if the motion be so slow that the squares and products of u, v, w and their first differential coefficients may be neglected, the equations (2) become is an exact differential. Hence, integrating, we see that ud.x + vdy+wdz consists of two parts, one of which is an exact differential, whilst the other does not contain t. In some cases, for example, when the motion is wholly periodic, we can assert that the latter part is zero, and therefore, that a velocity-potential exists. 25. Under the circumstances stated in Art. 23, the equations of Art. 6 are at once integrable throughout that portion of the mass for which a velocity-potential exists. For, in virtue of the reladv dw dw du du dv tions dz dy' dx dz' dy dx' equations in question may be written which are implied in (2), the d'o du dv dw dV 1 dp +u +v +w These have the common integral &c. &c. Here q denotes the resultant velocity √(u3 +v2 +w2), and F(t) is an arbitrary function of t, which may however be supposed included in do, since, by (2), the values of u, v, w are not thereby affected. аф whilst the equation of continuity ((9) of Art. 8) assumes the form In any problem to which these equations apply, and where the boundary-conditions are purely kinematical, the process of solution is as follows. We must first find a function & satisfying (5) and the given boundary-conditions; then substituting in (4) we get the value of p. Since the latter equation contains an arbitrary function of t, the complete determination of p requires a knowledge of its value at some point of the fluid for all values of t. 26. A comparison of equations (2) with the equations of Art. 15 gives a simple physical interpretation of the velocity-potential. Any actual state of motion of a liquid, for which a velocitypotential exists, could be produced instantaneously from rest by the application of a properly chosen system of impulsive pressures. This is evident from equations (21) Art. 15, which shew, moreover, that +const.; so that C-pp gives the requisite system. In = the same way app+ C gives the system of impulsive pressures which would completely stop the motion. The occurrence of an arbitrary constant in these expressions shews, what is otherwise evident, that a pressure uniform throughout a liquid mass produces no effect on its motion. In the case of a gas, is the potential of the external impulsive forces by which the actual motion at any instant could be produced instantaneously from rest. A state of motion for which a velocity-potential does not exist cannot be generated or destroyed by the action of impulsive pressures, or of external impulsive forces having a potential. 27. The existence of a velocity-potential indicates, besides, certain kinematical properties of the motion. A 'line of motion' is defined to be a line drawn from point to point, so that its direction is everywhere that of the motion of the fluid. The differential equations of the system of such lines The relations (2) shew that when a velocity-potential exists the lines of motion are everywhere perpendicular to a series of surfaces, viz. the surfaces = const. These are called the surfaces of equal velocity-potential, or more shortly, the equipotential surfaces. Again, if from the point (x, y, z) we draw a linear element ds in the direction (l, m, n), the velocity resolved in this direction is аф ах, аф ду, do dz dx ds dy ds dz ds' lu + mv + nw, or + + which do = ds The veloc ity in any direction is therefore equal to the rate of increase of in that direction. Taking ds in the direction of the normal to the surface = const. we see that if a series of such surfaces be drawn so that the difference between the values of p for two consecutive surfaces is constant and infinitely small, the velocity at any point will be inversely proportional to the distance between two consecutive surfaces in the neighbourhood of that point. Hence, if any equipotential surface intersect itself, the velocity is zero at every point of the intersection. The intersection of two distinct equipotential surfaces would imply an infinite velocity at all points of the intersection. Steady Motion. 28. When at every point the velocity at that point is constant in magnitude and direction, i. e. when everywhere, the motion is said to be 'steady.' ..(7) In steady motion the lines of motion coincide with the paths of the particles and are in this case called 'stream-lines.' For let P, Q be two consecutive points on a line of motion. A particle which is at any instant at P is moving in the direction of the tangent at P, and will, therefore, after an infinitely short time arrive at Q. The motion being steady, the lines of motion remain the same. Hence the direction of motion at Q is along the tangent to the same line of motion, i. e. the particle continues to describe the line of motion. In steady motion the equation (3) becomes dp = V-1q2+ constant........ (8). The equations of motion may however in this case be integrated to a certain extent without assuming the existence of a velocity-potential. For if ds denote an element of a stream-line, we have u = dx ds &c. Substituting in the equations of motion. This is of the same form as (8), but is more general in that it does not involve the assumption of the existence of a velocitypotential. It must however be carefully noticed that the 'constant' of equation (8) and the 'C' of equation (9) have very different meanings, the former being an absolute constant, while the latter is constant along any particular stream-line, but may vary as we pass from one stream-line to another. 29. The formula (9) may be deduced from the principle of energy without employing the equations of motion at all. Taking first the particular case of a liquid, let us consider the portion of an infinitely narrow tube, whose walls are formed of stream-lines, included between two cross sections A and B, the direction of motion being from A to B. Let p be the pressure, q the velocity, V the potential of the external forces, σ the area of the cross section at A, and let the values of the same quantities at B be distinguished by accents. In each unit of time a mass pqo at A enters the portion of the tube considered, whilst an equal mass pqo' leaves it at B. Hence qoq'o'. Again, the work done on the mass entering at A is pqo per unit time, whilst the loss of work at B is p'qo'. The former mass brings with it the energy pqo (q2+ V), whilst the latter carries off energy to the amount pq'o' (1⁄2 q'2+ V'). The motion being steady, the portion of the tube considered neither gains nor loses energy on the whole, so that 2 12 12 pqo +pqo (¿q2 + V) = p'q'o' + pq'o' (¿q′′2 + V'). Dividing by pqo (= pq'o'), we have |