CONTENTS. PAGE 1 17 On the Motion of a Piston and of the Air in a Cylinder On the Theories of the Internal Friction of Fluids in Motion, and of the Equilibrium and Motion of Elastic Solids SECTION I.-Explanation of the Theory of Fluid Motion proposed. Form- ation of the Differential Equations. Application of these Equations SECTION II. Objections to Lagrange's proof of the theorem that if udx+vdy+wdz is an exact differential at any one instant it is always the pressure being supposed equal in all directions. Principles of M. Cauchy's proof. A new proof of the theorem. A physical inter- pretation of the circumstance of the above expression being an exact SECTION III.-Application of a method analogous to that of Section I. to the determination of the equations of equilibrium and motion of SECTION IV.-Principles of Poisson's theory of elastic solids, and of the oblique pressures existing in fluids in motion. Objections to one of his hypotheses. Reflections on the constitution, and equations of motion of the luminiferous ether in vacuum On the Proof of the Proposition that (Mx + Ny)-1 is an Integrating Factor of On the Resistance of a Fluid to two Oscillating Spheres On the Critical Values of the Sums of Periodic Series SECTION I-Mode of ascertaining the nature of the discontinuity of a function which is expanded in a series of sines or cosines, and of obtaining the developments of the derived functions . SECTION II.-Mode of ascertaining the nature of the discontinuity of the integrals which are analogous to the series considered in Section I., and of obtaining the developments of the derivatives of the SECTION III.-On the discontinuity of the sums of infinite series, and of the values of integrals taken between infinite limits SECTION IV.-Examples of the application of the formulæ proved in the ERRATA. P. 103, 1. 14, for their read there. 31 MATHEMATICAL AND PHYSICAL PAPERS. [From the Transactions of the Cambridge Philosophical Society, ON THE STEADY MOTION OF INCOMPRESSIBLE FLUIDS. [Read April 25, 1842.] In this paper I shall consider chiefly the steady motion of fluids in two dimensions. As however in the more general case of motion in three dimensions, as well as in this, the calculation is simplified when uda+vdy+wdz is an exact differential, I shall first consider a class of cases where this is true. I need not explain the notation, except where it may be new, or liable to be mistaken. To prove that uda+vdy+wdz is an exact differential, in the case of steady motion, when the lines of motion are open curves, and when the fluid in motion has come from an expanse of fluid of indefinite extent, and where, at an indefinite distance, the velocity is indefinitely small, and the pressure indefinitely near to what it would be if there were no motion. By integrating along a line of motion, it is well known that we get the equation P = V − } (u2 + v2 + w3) + С.......... P .(1), where dV=Xdx + Ydy + Zdz, which I suppose an exact differential. Now from the way in which this equation is obtained, it appears that C need only be constant for the same line of motion, and therefore in general will be a function of the parameter of a line of motion. I shall first shew that in the case considered C is absolutely constant, and then that whenever it is, uda+vdy+wdz is an exact differential *. To determine the value of C for any particular line of motion, it is sufficient to know the values of p, and of the whole velocity, at any point along that line. Now if there were no motion we should have P, being the pressure in that case. But considering a point in this line at an indefinite distance in the expanse, the value of p at that point will be indefinitely nearly equal to p1, and the velocity will be indefinitely small. Consequently C is more nearly equal to C, than any assignable quantity: therefore C is equal to C1; and this whatever be the line of motion considered; therefore C is constant. In ordinary cases of steady motion, when the fluid flows in open curves, it does come from such an expanse of fluid. It is conceivable that there should be only a canal of fluid in this expanse in motion, the rest being at rest, in which case the velocity at an infinite distance might not be indefinitely small. But experiment shews that this is not the case, but that the fluid flows in from all sides. Consequently at an indefinite distance the velocity is indefinitely small, and it seems evident that in that case the pressure must be indefinitely near to what it would be if there were no motion. Differentiating therefore (1) with respect to x, we get and therefore udx + vdy+wdz is an exact differential. When uda+vdy+wdz is an exact differential, equation (1) may be deduced in another way †, from which it appears that Cis constant. Consequently, in any case, udx + vdy+wdz is, or is not, an exact differential, according as C is, or is not, constant. Steady Motion in Two Dimensions. I shall first consider the more simple case, where udx + vdy is an exact differential. In this case u and v are given by the equations du dv + [This conclusion involves an oversight (see Transactions, p. 465) since the three preceding equations are not independent, as may readily be seen. I have not thought it necessary to re-write this portion of the paper, since in the two classes of steady motion to which the paper relates, namely those of motion in two dimensions, and of motion symmetrical about an axis, the three analogous equations are reduced to one, and the proposition is true. None of the succeeding results are affected by this error, excepting that the second paragraph of p. 11 must be restricted to the two cases above mentioned.] + See Poisson, Traité de Mécanique. |