gral in question will still vanish. On this understanding then the relation (8) is satisfied, and the values of u, v, w thus derived (which we shall now distinguish by a new suffix) will satisfy (3) and (4). They will not however in general satisfy the boundary-condition (5). Let λ be the value of the normal velocity which the formulæ (6) would give, viz. and let us write λ = lu+mv。 +nw。, u = u2+ u2, v = v2 + v1, w = w。 + w1, 0 where u1, v1, w1 remain to be found. Substituting in (3), (4) and (5) we obtain 1 dw. 1 dz du, dw, dx = = throughout the (simply-connected) region, and making .(17) .(18) at the boundary. The problem of finding Q so as to satisfy these conditions was shewn in Art. 49 to be determinate. Vortex-sheets. 132. We have so far assumed u, v, w to be continuous. We will now shew how cases where these functions are discontinuous may be brought within the scope of our theorems. Let us suppose that we have a series of vortex-filaments arranged in a thin film over a surface S, and let w be the angular velocity, and e the thickness, at any point of such a film. Let us examine the form which our previous results assume when w is increased, and e diminished, without limit, yet in such a way that the product we, (=w', say,) remains finite. The infinitely thin film is then called a ' vortex-sheet.' The functions L, M, N will now consist in part of potentials of matter distributed with surface densities ζε ξε ηε over S. We know from the theory of Attractions that L, M, N are continuous even when the point to which they refer crosses S, but that their derivatives are discontinuous; viz. the derivative taken in the direction of the normal (drawn in the direction of crossing) experiences an abrupt decrease of amount 47 × surface-density. dL dL dL Hence the changes (diminutions) in the values of dx' dy dz will be 2le, 2m§e, 2nge, if l, m, n be the direction-cosines of the normal drawn as just explained. The values of u, v, w obtained from (6) will therefore be discontinuous at S, the components of the relative velocity of the portions of fluid on opposite sides of S being 2 (my — nn) e, 2(ng – Ì¿) e, 2(ln – m§)e................(19), respectively. These are the amounts by which the components on the side towards which the normal (l, m, n) is drawn fall short of those on the other. This relative velocity is tangential to S, and perpendicular to the vortex-lines. Its amount is 2we, or 2w', and its direction is that due to a rotation of the same sign as o' about the vortex-lines in the adjacent part of S. Hence a surface of discontinuity at which the relation lu1 + mv1 + nw1 = lu2+mv2+ nw,... 2 2 (20), [(13) of Art. 10] is satisfied may be treated as a vortex-sheet, in which the vortex-lines are everywhere perpendicular to the direction of relative motion of the fluid on the two sides of the surface, and the product w' of the (infinite) angular velocity into the (infinitely small) thickness is equal to half the amount of this relative velocity. In the same way, a discontinuity of normal velocity is obtained. by supposing to be infinite throughout a thin film, but in such a way that the product (e' say) of 0 into the thickness e is finite. The normal velocities at adjacent points on opposite sides of the film will then differ by O'. Velocity-Potential due to a Vortex. 133. At points external to the vortices there exists of course a velocity-potential, whose value may be found by integration of (15), as follows. Taking, for shortness, the case of a single closed vortex, we write da'dy'dz' = o'ds', where ds' is an element of the length of the filament, σ' its section. Also we may write where the product w'o', the strength of the vortex, being constant, is placed outside the sign of integration, which is taken right round the filament. Now the analytical theorem (7) of Art. 40 enables us to replace a line-integral taken round a closed curve by a surface-integral taken over any surface bounded by that curve. To apply this to our case, we write, in the formula cited, Here l, m, n denote the normal to the element dS' of a surface bounded by the vortex-filament. The equation (22) may be otherwise written denotes the angle between r and the normal l, m, n. where cos Since .ds' is the elementary solid angle subtended by dS' at (x, y, z), we see that the velocity-potential at any point due to w'o' into 2π a single re-entrant vortex is equal to the product of the solid angle which any surface bounded by the vortex subtends at that point. Since this solid angle changes by 4 when the point in question describes a circuit embracing the vortex, the value of given by (23) is cyclic, the cyclic constant being twice the strength of the vortex. Compare Art. 127. Dynamical Theorems. 134. In the theorems which follow, we assume that the external impressed forces have a single-valued potential V, and that p is either a constant or a function of p only. We first consider any terminated line AB drawn in the fluid, and suppose every point of this line to move with the velocity of the fluid at that point. In other words the line moves so as to consist always of the same chain of particles. We proceed to calculate the rate at which the flow along this line, from A to B, is increasing. If dx, dy, dz be the projections on the axes of co-ordinates of an element of the line, we have, with our previous notation, ǝdx Now the rate at which do is increasing in consequence of at the motion of the fluid, is evidently equal to the difference of the velocities parallel to x at its two ends, i.e. to du; and the Integrating along the line, from A to B, we get or, the rate at which the flow from A to B is increasing is equal to the excess of the value which 192· which it has at A. dp ·V. has at B over that ρ This theorem, which is due to Thomson, comprehends the whole of the dynamics of a perfect fluid in the general case, as equation (3) of Art. 25 does for the particular case of irrotational motion, For instance, equations (26) of Chapter 1. may be derived from it by taking as the line AB the infinitely short line whose projections were originally da, db, dc, and equating separately to zero the coefficients of these infinitesimals. The expression within brackets on the right-hand side of (24) is a single-valued function of x, y, z. It follows that if the integration on the left-hand side be taken round a closed curve, (so that B coincides with A,) we have or, the circulation in any circuit moving with the fluid does not alter with the time. See Art. 59. Applying this theorem to a circuit embracing a vortex-tube we find that the strength of any vortex is constant. Also, remembering the formula given in Art. 39 for the circulation in an infinitesimal circuit, we see that if throughout any |