whence we easily see that μr is one such factor. = The equation which U is to satisfy will be got by expressing s and w in terms of U, and substituting in (19) in the general case, or by substituting in (18), in the case where udx+vdy+wdz is an exact differential. In the latter case the equation which U is to satisfy is In the general case, the equation is what I shall write dU (1/d2 U d2U 1dU d U d + = 0 dr2 ... r dr (21). ds = dz = { d (s2 + w2) + (wdr-sdz)+ (sdz - wdr) dw dr Hence the quantity under the integral sign must be a function of U. And in fact, we can easily shew by trial that is a first integral of (21). The last term of (22) is the value of the constant in (1). By expanding U in a series ascending according to integral powers of z, which may be done as long as the origin is arbitrary, it will be found that the integral of (20) may be written under the form We may employ equations (21) or (20) just as before, to determine whether the motion in a proposed system of lines is possible. If F(r, z) = U1 = C be the equation to the system, we must have, as before, U≈ 4 (U1); whence we get, in the general case, 1 2 +'" (U) { (U d - U d) [; (U)2 + (W)*"]} dz dr dz and in the more restricted case where udx + vdy+wdz is an exact As before, the ratio of the coefficients of p′′ (U1) and p′ (U) must be a function of U, alone, when z, r and U, are connected by the equation F(r, z) = U1. If the motion be possible, it will in general be determinate, U being of the form Aƒ (r, z) + B. If U1=r however, the form of remains arbitrary. In this case the fluid may be conceived to move in cylindrical shells parallel to the axis, the velocity being any function of the distance from the axis. 1 Particular cases are, where the lines of motion are right lines directed to a point in the axis, and where they are equal parabolas having the axis of z for a common axis. In these cases is an exact differential. udx+vdy+wdz We may employ equations (20) and (21) to determine whether the hypothesis of parallel sections can be strictly true in any case. In this case, the sections being perpendicular to the axis of z, we must have Substituting this value in (21), we find, by equating to zero coefficients of different powers of r, that the most general case corresponds to U= (a + bz + cz3) r2 + ez +ƒ. If udx+vdy+wdz be an exact differential, the most general case corresponds to U = (a + bz) r2 + c + ez. [From the Transactions of the Cambridge Philosophical Society, ON SOME CASES OF FLUID MOTION. [Read May 29, 1843.] THE equations of Hydrostatics are founded on the principles that the mutual action of two adjacent elements of a fluid is normal to the surface which separates them, and that the pressure is equal in all directions. The latter of these is a necessary consequence of the former, as has been shewn by Mr Airy*. An exactly similar proof may be employed in Hydrodynamics, by which it may be shewn that, if the mutual action of two adjacent elements of a fluid in motion is normal to their common surface, the pressure must be equal in all directions, in order that the accelerating force which acts on the centre of gravity of an element may not become infinite, when we suppose the dimensions of the element indefinitely diminished. In Hydrostatics, the accurate agreement of the results of our calculations with experiments, (those phenomena which depend on capillary attraction being excepted), fully justifies our fundamental assumption. The same assumption is made in Hydrodynamics, and from it are deduced the fundamental equations of fluid motion. But the verification of our fundamental law in the case of a fluid at rest, does not at all prove it to be true in the case of a fluid in motion, except in the very limited case of a fluid moving as if it were solid. Thus, oil is sufficiently fluid to obey the laws of fluid equilibrium, (at least to a great extent), yet no one would suppose that oil in motion ought to be considered a perfect fluid. It would appear from the following consideration, that the fluidity of water and other such fluids is not quite perfect. * See also Professor Miller's Hydrostatics, page 2. When a mass of water contained in a vessel of the form of a solid of revolution is stirred round, and then left to itself, it presently comes to rest. This, no doubt, is owing to the friction against the sides of the vessel. But if the fluidity of water were perfect, it does not appear how the retardation due to this friction could be transmitted through the mass. It would appear that in that case a thin film of fluid close to the sides of the vessel would remain at rest, the remaining part of the fluid being unaffected by it. And in this respect, that part of Poisson's solution of the problem of an oscillating sphere, which relates to friction, appears to me in some degree unsatisfactory. A term enters into the equation of motion of the sphere depending on the friction of the fluid on the sphere, while no such term enters into the equations of motion of the fluid, to express the equal and opposite friction of the sphere on the fluid. In fact, as long as we regard the fluidity of the fluid as perfect, no such term can enter. The only way by which to estimate the extent to which the imperfect fluidity of fluids may modify the laws of their motion, without making any hypothesis as to the molecular constitution of fluids, appears to be, to calculate according to the hypothesis of perfect fluidity some cases of fluid motion, which are of such a nature as to be capable of being accurately compared with experiment. The cases of that nature which have hitherto been calculated, are by no means numerous. My object in the present paper which I have the honour to lay before the Society, has been partly to calculate some such cases which may be useful in determining how far we are justified in regarding fluids as perfectly fluid, and partly to give examples of the methods by which the solution of problems depending on partial differential equations may be effected. In the first seven articles, I have mentioned and explained some general principles, which are afterwards applied. Some of these are not new, but it was convenient to state them for the sake of reference. Others are I believe new, at least in their development. In the remaining articles, I have given different problems, of which I have succeeded in obtaining the solutions. As the problem to be solved is usually stated at the head of each article, I shall here only mention some of the results. As a particular case of the problem given in Art. 8, I find that, when a cylinder oscillates in an infinitely extended fluid, the effect of the inertia of the fluid is to increase the mass of the cylinder by that of |