independent of the time t. Let, for the sake of brevity, dt T dt T dt (11c) equation (14) was given by Maxwell in 1867. Using these abbreviations, equations (9) and (10) may be written :— (15) These equations have, in our theory, the same significance as that which, in the classical theory*, is possessed by the known equations which give the quantities (Prx-p) &c. as functions of the components e, f, g, a, b, c of the velocity of deformation. They contain the terms Ce-/T &c., which do not appear in the ordinary equations. Further, in these equations, the functions E, F, G, A, B, C, O, defined by (11), *Stokes, Mathematical and Physical Papers,' vol. i. p. 90, eq. (8); Cambridge, 1880. Basset, 'A Treatise on Hydrodynamics, vol. ii. p. 241, eq. (16); Cambridge, 1888. Lamb, 'Hydrodynamics,' p. 512, eq. (4) & (5); Cambridge, 1895. (12), and (13), take the place occupied in the ordinary equations by the components e, f, g, a, b, c, w, and play precisely the same part. § 9. The constants λ and μ, defined by equations (15) and (14) of the preceding paragraph, are the two coefficients of viscosity in our theory. Authors who have dealt with the problem of viscosity have generally used two constants which they have often denoted by the same symbols: A, u. Poisson, in the memoir already quoted *, introduces two constants which, at least in the general case, are independent of each other. In 1843, Barré de Saint-Venant †, and in 1845, with clearness and precision, Sir G. G. Stokes ‡, pointed out the considerations which led to the conclusion that This is the relation arrived at in 1867 by Clerk-Maxwell §, who was guided by the kinetic theory. We find the same relation adopted by Kirchhoff |, Basset ¶, Lamb **, and many other writers; nevertheless, in 1874, O. E. Meyer †† proposed a totally different relation. Lastly, Voigt ‡‡ has quite recently called into question the existence of any relation whatever connecting the two constants of viscosity. One may hope to find some indications regarding the value of the ratio / by calculating, for a fluid, the quantity (already considered by Stokes and by Helmholtz) which has, after Lord Rayleigh, been called the dissipation function. This is the method which was suggested, apparently for the first time, by Jacobi in the theory of elasticity; it has been followed by Duhem §§ in the theory of viscosity of fluids. In this latter case there undoubtedly exists a dissipation function, and it is always positive. This condition is easily * Journal de l'Ecole Polytechnique, 20 Cahier, tome xiii. (1831). + Comptes Rendus, tome xvii. p. 1240 (1843). 6 Transactions of the Cambridge Philosophical Society, vol. viii. p. 287 (1845); Mathematical and Physical Papers,' vol. i. p. 75 (1880); see $$ 3, 4, & 18. § Philosophical Transactions, vol. clvii. pp. 81–82 (1867). Scientific Papers, vol. ii. p. 69 (1890). Vorlesungen über die Theorie der Wärme, p. 193 (1894). A Treatise on Hydrodynamics,' vol. ii. p. 242 (1888). **Hydrodynamics,' p. 512 (1895). ++ Crelle's Journal f. reine u. angew. Mathematik, Bd. lxxviii. p. 130 (1874). Kinetische Theorie der Gase, II. Auflage, Mathem. Zusätze, pp. 112-114 (1899). Kompendium d. theoretischen Physik, Bd. i. p. 462 (1895). $$ Théorie thermodynamique de la viscosité, du frottement et des faux équilibres chimiques, Paris 1896, p. 52. shown to be equivalent to provided λ> -μ, we suppose μ > 0, which is legitimate. (2) Let us now return to the study of the two constants of viscosity introduced into our theory by means of equations (14) and (15) of the preceding paragraph. These equations teach us that the relation λ=, proposed by Stokes and accepted by the majority of scientists, is an immediate consequence of the equation hk, of which we have given a detailed discussion above. If, on the other hand, we suppose that h and k may be unequal, then and the relation connecting λ and u will depend not only on the ratio k/n, but also on that of the new constant h to the rigidity n. As to the inequality (2), the only consequence which may be deduced from it is that and this new inequality is certainly verified in the case of actual fluids. In conclusion, we may say that the equation hk, and Stokes's relation, λ=-3, agree perfectly with the whole of our hypotheses; but there is nothing to lead us to regard them as necessary corollaries of our theory. §10. Let X, Y, Z be the components, per unit of mass, of the external force which acts on an element of volume at the point (x, y, z). Then we have three equations, the first of which is From this, taking into account equations (16) and (17) of § 8, In this equation the symbol V2 represents the well-known operator of Laplace, and o has the meaning which we have assigned to it in § 1; similarly two analogous equations are found. These three equations are obviously the equations of motion of the fluid, which it was our object to study. In a future contribution we hope to be able to give some applications of the theory which we have developed. IN XXXV. On the Breaking of Waves. By W. G. FRASER, M.A., Queen's College, Cambridge*. N the ordinary theory of the reflexion of waves at a fixed barrier, it is shown that an incident train of waves gives rise to a reflected train of the same type and amplitude, the two trains combining to form a system of standing waves. Now the theory claims to be no more than a first approximation, applicable only to small disturbances; but it is a matter of everyday knowledge that waves which, if left to themselves, would proceed for a considerable distance without sensible change of type, may nevertheless be too high to be reflected at a wall according to the ordinary theory; instead of this, they break into spray against the wall. Thus it appears that the theory of progressive waves is a better approximation to fact than the theory of reflected waves. An attempt is here made to find part, at least, of the reason for this in the friction of the wall. It appears that, in deep water, if the ratio of the amplitude to the wave-length exceeds a certain small amount, the wave will break; in shallow water the breaking amplitude is somewhat less than is indicated by this ratio. In the ordinary theory, if the incident train of waves have a velocity potential we ascribe to the reflected train a potential p', so adjusted as to bring the water in contact with the barrier into a state of motion that is purely vertical; so that, if u, v be the horizontal and vertical components of the velocity of an element of liquid close to the barrier due to the incident wave, the components due to the reflected wave are u, v. The two velocity systems therefore bear the same relation to one another as the velocity of a particle before and after impact at a smooth vertical elastic wall. Now, while the horizontal motion of the liquid in contact with the wall must be annihilated, it is only natural to suppose that the wall exerts an appreciable drag on the vertical motion. It is proposed, therefore, to take, as the velocities in the reflected wave -u, mv, where m is slightly less than *Communicated by the Author. nity. This motion is inconsistent with the equation of continuity of an incompressible fluid, and if set up must lead to instant rupture of the continuity of the liquid, which will break up into drops. But liquids appear to be capable of resisting a small amount of internal tension, and this, together with the surface-tension, may suffice, when the motion is not too violent, to prevent the rupture, modifying the motion in such a way as to render it consistent with the equation of continuity. If m does not differ much from unity, the difference between this actual motion and the motion given by the ordinary theory on the one hand, and that obtained on the hypothesis of compressibility on the other hand, will be small compared with the motion itself. If then we substitute the velocities -u, mv in the equation of continuity for a compressible fluid, the variation of density may be taken as an indication of the tendency to rupture, and we may say that rupture will take place if the maximum density exceeds, or the minimum density falls short of the actual density of the liquid by more than a certain small amount depending on the nature of the liquid. Deep Water Waves. Taking the origin at the obstacle, and measuring y vertically downwards, let the incident waves have a profile given by n=sin k(x-ct) indicating a train of waves of altitude 7 and length 2π/k advancing parallel to the axis of æ. The velocity potential for such a train is so that the velocity system at any point of the reflected wave is u-cke-ky sin k(x + ct); v'=—cke-ky cos k(x+ct). In accordance with our hypothesis, therefore, we investigate the system u-cke-ky sin k(x+ct); -mcke-ky cos k (x+ct), and the motion of any element is supposed to be compounded of this and the motion due to the incident wave. Phil. Mag. S. 6. Vol. 2. No. 10. Oct. 1901. The latter 2 B |