In this case p, u and v, are given by the equations

dy

We still have

dx U

for the differential equation to a line of

motion, where udy - vdx is still an exact differential, on account of equation (9). Eliminating p by differentiation from (7) and (8), and expressing the result in terms of U, we get the equation which U is to satisfy, viz.

*

[This equation may be applied to prove an elegant theorem due to Mr F. D. Thomson {see the Oxford, Cambridge, and Dublin Messenger of Mathematics, Vol. III. (1866), p. 238, and Vol. iv. p. 37}, that if a vessel bounded by a cylindrical surface of any kind and by two planes perpendicular to its generating lines be filled with homogeneous liquid, and the whole be revolving uniformly about a fixed axis parallel to its generating lines, then if the vessel be suddenly arrested the motion of the liquid will be steady.

If w be the angular velocity, we shall have for the motion before impact

U - f(wy dy + wx dx) = − & w (x2 + y2) = − ¿ wr2,

omitting the constant as unnecessary. If u, v be the components of the change of velocity produced by impact, it follows from the equations of impulsive motion that udx+vdy will be a perfect differential dp, where p satisfies the partial differential d2 d2 equation y=0, v standing for + If U' be the U-function corresponding dx2TM dy2*

to this motion—and such a function exists by virtue of the equation of continuity whether the motion be steady or not-we have

where the quantity under the sign fis is a perfect differential by virtue of the equa

tion yo=0; and we see at once that U'=0. Hence for the whole motion just after impact

v(U+U')=yU-2w,

which satisfies the equation of steady motion (10); and as the condition at the boundary, namely that the fluid shall slide along it, is satisfied, being satisfied initially, it follows that the initial motion after impact will be continued as steady motion.

To actually determine the function & or U', and thereby the motion in any given case, we must satisfy not only the general equation 76=0 but also the equation of condition at the boundary, namely that there shall be no velocity in a direction normal to the surface, which gives

аф dx

wy) dy - (2

аф dy

wx) dx=0......

...(a),

at any point of the boundary. If ƒ (x, y) =0 be the equation of the boundary, we must substitute - df/dxdf/dy for dy/dx in (a), and the resulting equation will have to be satisfied when f=0 is satisfied.

There are but few forms of boundary for which the solution of the problem can be actually effected analytically, among which may be mentioned in particular the case of a rectangle. But by taking particular solutions of the equation y = 0, substituting in (a) and integrating, which gives

or what comes to the same thing taking particular solutions of the equation [U'=0 and substituting in (B), which gives the general equation of the lines of motion, we may synthetically obtain an infinity of examples in which the conditions of the problem are satisfied, any one of the lines of motion being taken as the boundary of the fluid.

Thus for Ukr3 cos 30 we have for the lines of motion

which therefore are cubic curves, recurring when is increased by 120o. (8) is satisfied by

Hence when k has the above value the cubic curve (7) breaks up, for the particular value of the parameter C above written, into three straight lines forming the sides of an equilateral triangle, and the vessel may therefore be supposed to be an equilateral triangular prism. The various lines of motion correspond to values of the parameter C from 0 to - wa2. This case is given by Mr Thomson.

U'=kr2 cos 20 leads to the case of steady motion in similar and concentric ellipses considered in the text a little further on, which therefore may be conceived to have been produced from motion about a fixed axis as pointed out by Mr Thomson. In fact, any case of steady motion in two dimensions in which U=const. may be conceived to have been so produced.]

of (10). Consequently this latter term, which is the value of C in (1), comes out a function of the parameter of a line of motion as it should.

We may employ equation (10), precisely as before, to enquire whether a proposed system of lines can, under any circumstances, be a system of lines of motion. Let f (x, y) = U1 = C, be the equation to the system; then, putting as before, U=$(U1), we get

or,

1

dx dx dy

1

1

1

+ 4' (U) (dy, ddl, d) (du + )=0;

Pp′′ (U1) + Q&′ (U1) = 0, suppose.

U

d2 U

dy2

Hence, as before, if we express y in terms of x and U1, from Q the equation f(x, y) = U1, and substitute that value in the P p'

result must not contain x. If it does, the proposed system of lines cannot be a system of lines of motion; if not, the integration of the above equation will give & (U1), under the form

$ (U1) = AF (U1) + B,

and we can immediately get the values of u, v and p, with the same arbitrary constants as in the previous case.

One case in which the motion is possible is where the lines of motion are a system of similar ellipses or hyperbolas about the same centre, or a system of equal parabolas having the same axis. In the case of the ellipse, the particles in a radius vector at any time remain in a radius vector, and the value of p has the form

pV + A + B (x2 + y2).

When however the ellipse becomes a circle, P and Q vanish in the equation Po′′ (U12)+Qp' (U1) = 0. Consequently the form of may be any whatever. The value of U, being x2 + y2, we have

24′ (U1) y, v=-
- 24' (U1) x;

2

whence, u2 + v2 = 4 {p′ ( U,)}2 (x2 + y2) = 4U1 {6′ (U1)}2.

1

Hence, the velocity may be any function of the distance from the centre. It is evident that we may conceive cylindrical shells of fluid, having a common axis, to be revolving about that axis with any velocities whatever, if we do not consider friction, or whether such a mode of motion would be stable. The result is the same if we enquire in what way fluid can move in a system of parallel lines.

In any case where the motion in a certain system of lines is possible, if we suppose two of these lines to be the bases of bounding cylindrical surfaces, and if we suppose the velocity and direction of motion, at each point of a section of the entering, and also of the issuing fluid, to be what that case requires, I have not proved that the fluid must move in that system of lines. When the above conditions are given there may still perhaps be different modes of steady motion; and of these some may be stable, and others unstable. There may even be no stable steady mode of

motion possible, in which case the fluid would continue perpetually eddying.

In the case of rectangular hyperbolas, the fluid appeared, on making the experiment, to move in hyperbolas when the end at which the fluid entered was broad and the other end narrow, but not when the end by which the fluid entered was narrow. This may, I think, in some measure be accounted for. Suppose fluid to flow out of a vessel where the pressure is p, into one where it is p1, through a small orifice. Then, the motion being steady, we have, along the same line of motion, p/p = C — v2, where v is the whole velocity. At a distance from the orifice, in the first vessel, the pressure will be approximately p1, and the velocity nothing. At a distance in the second vessel, the pressure will 2 (P、 — P2),

be approximately p2, and therefore the velocity

1

ρ

nearly. The result is the same if forces act on the fluid. Hence the velocity must be approximately constant; and therefore, the fluid which came from the first vessel, instead of spreading out, must keep to a canal of its own of uniform breadth. This is found to agree with experiment. Hence we might expect that in the case of the hyperbolas, if the end at which the fluid entered were narrow, the entering fluid would have a tendency to keep to a canal of its own, instead of spreading out.

In ordinary cases of steady motion, when the lines of motion. are open curves, the fluid is supplied from an expanse of fluid, and consequently udx+vdy+wdz is an exact differential. Consequently, cases of open curves for which it is not an exact differen- 3 tial do not ordinarily occur. We may, however, conceive such cases to occur; for we may suppose the velocity and direction of motion, at each point of a section of the entering, and also of the issuing stream, to be such as any case requires, by supposing the fluid sent in and drawn out with the requisite velocity and in the requisite direction through an infinite number of infinitely small tubes.

In the case of closed curves however, in whatever manner the fluid may have been put in motion, it seems probable that, if we neglect the friction against the sides of the vessel, the fluid will have a tendency to settle down into some steady mode of motion. Consequently, taking account of the friction against the sides of