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 %, 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); whence we get, in the general case, U = = dU, dr dz d2 U1 + p (U) { (d) d - du; d) [ (10 {(dz' d2 U1 + = 0, dr dr dz2 dr2 r dr and in the more restricted case where udx + vdy + wds is an exact differential, we get d2 U1 U dz 4" (U) {(d-')' + (dv)} + •'(U) (AU) + QU - 1 dl) dr2 = 0. r dr 1 As before, the ratio of the coefficients of p′′ (U1) and p′(U1) must be a function of U1 alone, when ≈, r and U1 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 Ur 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. 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 for a common axis. In these cases udx + vdy + wdz is an exact differential. 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 ≈, 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 + b≈ + cx2) r2 + ez +f. If udx + vdy + wdz be an exact differential, the most general case corresponds to XXIV. On the Truth of the Hydrodynamical Theorem, that if udx +vdy+wdz be a Complete Differential with respect to x, y, z, at any one instant, it is always so. By the Rev. J. Power, M.A., Fellow and Tutor of Trinity Hall. [Read May 9, 1842.] THIS Theorem was first announced by La Grange, who has given a demonstration of it in the Mécanique Analytique, Tom. II. p. 307. The late celebrated mathematician, Baron Poisson, has, however, in the last edition of his Mechanics, expressed great doubts of its generality, and has even mentioned that examples have occurred to him in which it is in fault. Those examples, however, he has not given, which is much to be regretted, as the theorem is one of the greatest importance in the theory of fluid motion, and if not generally true, it was highly desirable for the prevention of error, that its want of generality should be placed beyond all doubt, which a single legitimate exception would have been sufficient to effect. The demonstration of La Grange supposes that the general values u, v, w, the component velocities of any given particle of fluid at the end of the time t, are developable as follows: u = u' + u" t + u"" t2 + &c. v = v′ + v′′ t + v"" ť2 + &c. w = w' + w" t + w'" ť2 + &c. and Poisson objects that the demonstration fails when u, v, w are not developable in series of the above form, as may occasionally happen. The objection is a fair and reasonable one; and it is my object in |