portion of a fluid mass in motion the conditions = 0, n = 0, 5=0 obtain at any one instant, the same is true for the same portion of the mass at every other instant, which is the theorem of Art. 23. It follows that rotational motion cannot be produced in any part of a fluid mass by the action of forces which have a singlevalued potential, and that such a motion, if already existent, cannot be destroyed by the action of such forces. If we take at any instant a surface composed wholly of vortexlines, the circulation in any circuit drawn on it is zero, by Art. 40, for we have l§+my+n§= 0 at every point of the surface. The preceding article shews that if the surface be now supposed to move with the fluid, the circulation will always be zero in any circuit drawn on it, and therefore the surface will always consist of vortex-lines. Again, considering two such surfaces, it is plain that their intersection must always be a vortex-line, whence we derive the theorem that the vortex-lines move with the fluid. This remarkable theorem was first given by Helmholtz* for the case of liquids; the preceding proof, by Thomson, shews it to be applicable to all fluids satisfying the conditions stated at the beginning of this article. Kinetic Energy. 135. The formula for the kinetic energy, viz. 2T= SSS p(u2 + v2 + w2) dx dy dz.. .(25), may be put into several remarkable and useful forms. We confine ourselves, for simplicity, to the case where the fluid (supposed incompressible) extends to infinity and is at rest there, and where further all the vortices present are within a finite distance of the origin. We have in this case, 0=0, P=0, p = const., so that (25) becomes on substitution from (6), This triple integral may, exactly as in Art. 129 (12), be replaced by the sum of a surface-integral pSS{L (nv − mw) + M (lw − nu) + N (mu — lv)} dS.............. (26), and a volume-integral dw L du M dz du ...... • SSS {I (dy − de) + 1 (de − de) + N (de – dy)} dadydz P dz Now it appears from (11) that at an infinite distance R from the 1 origin, L, M, N are at most* of the order and therefore u, v, ข R' 1 at most of the order whereas when the external bounding surR2, face is increased in all its dimensions without limit the surfaceelements dS increase proportionately to R2 only. The surface1 integral (26) is therefore of an order not higher than and therefore vanishes in the limit. Hence ..... ...... R' T=pSSS (L§+Mn+NÇ) dx dydz.......................... .(28). If we substitute the values of L, M, N from (11), this becomes where each of the volume integrations extends over all the vortices. 136. Under the same circumstances we have another useful expression for T; viz. T = 2pƒfƒ{u (y5 — zn) + v(z§ − x5)+w(xn− y§)}dx dydz.....(30). To verify this, we take the right-hand member, and transform it by the process already so often employed, omitting the surfaceintegrals for the same reason as in the preceding article. The first of the three terms gives 172/2 as may be seen (for example) by calculating the value of L for a single closed vortex, and expressing it, by the method of Art. 133, as a surface-integral taken over a surface bounded by the vortex. Conse Transforming the remaining terms in the same way, adding, and making use of the equation of continuity or, finally, on again transforming the last three terms, i.e. T. 1⁄2p SSS (u2 + v2 + w2) dx dy dz, The value (30) of T must of course be unaltered by any displacement of the axes of co-ordinates. This consideration gives (wn-v) dx dydz = 0 fff (v — un) dx dydz= 0 (31), relations which of course admit of independent verification. Thus 137. The rate at which the energy of any mass of liquid is increasing at any instant is if l, m, n be the direction cosines of the inwardly-directed normal to any element dS of the boundary. The part of this expression which contains p gives the rate at which the external pressure works, the remaining part expresses the rate at which the mass is losing potential energy. If the mass be enclosed within fixed rigid walls, we have lu + mv + nw = = 0 at the boundary, and therefore dT 0, or T: = const. The same result holds for the case of an unlimited mass of liquid subject to the conditions of Art. 129. We then have, beyond the vortices and it appears from Chapter III. that at an infinite distance from аф dt the origin, and therefore also is constant with respect to x, y, z. Under these circumstances the surface integral in (32) is zero. Compare Art. 65. 138. We proceed to apply the foregoing general theory to the discussion of some simple cases. 1. Rectilinear Vortices. Suppose that we have an infinite mass of liquid in motion in two dimensions (xy), so that u, v are functions of x, y only, and w=0. We have then = 0, 70 everywhere and therefore also L=0, M=0. The value of N is η = and if we perform the integration with respect to z' between the limits, and then make y infinite, we find where r now (and as far as Art. 140) stands for {(x − x')2 + (y — y')2}1. The first term in the value of N, though infinite, is constant, and since we are concerned only with the differential coefficients of N, A vortex-filament whose co-ordinates are ', y' and strength m' contributes to the motion at (x, y) a velocity whose components This velocity is perpendicular to the line joining the points (x, y) Let us calculate the integrals ffuçdxdy, and ssvçdady, where the integrations include all portions of the plane xy for which does not vanish. We have where each double integration includes the sections of all the vortices. Now, corresponding to any term |
