SURFACE INTEGRAL.

201

where x is one of the small pieces, and w the spin at some point within it. The approximation may be made as close as we like by taking the pieces small enough, and therefore the circulation is exactly equal to the integral 2fSwdx. If dλ be a small piece of the contour abcd, we know that the circulation is also equal to - Sodλ; and consequently we have 2 Swdx = fSodλ.

In general, if is any vector having a definite value at every point of space, the integral -JSwda, taken over any area, plane or curved, is called the surface-integral of the vector over that area. We may therefore state our proposition thus: the line-integral of the velocity round any contour is equal to twice the surface-integral of the spin over any cap covering the contour.

d

Let us now draw another cap, agc, covering the contour. Then the surface-integral of the spin over agc must be equal to that over aec, because each is half the line-integral of velocity round abcd. But in one case the vectors representing small pieces of area will all be drawn inwards, and in the other outwards. If then we suppose them all to be drawn outwards, the surface-integral over the entire closed surface aecg will be zero. It is in fact obvious that if we divide the area of any closed surface into small pieces, and suppose each of these to be gone round in a counter-clockwise direction, as viewed from outside, the sum of all their circulations will be zero, since each boundary line is traversed twice, in opposite directions.

We learn, therefore, that the surface-integral of the spin over any closed surface is zero. The closed surface may be that of a body having no holes through it, as in the figure, or it may be that of a body with any number of holes through it; for example, the surface of an anchorring, or of a solid figure-of-eight.

Let us now apply this proposition to a portion of a tube of spin, cut off at a and b by surfaces of any form.

This closed surface consists of the two ends at a and b, and of the tubular portion between them. At every point of the tubular portion the axis of spin is tangent to the vortex line through that point, which lies entirely in the surface; consequently it has no component normal to the surface. Therefore the tubular portion of the surface contributes nothing to the surface-integral. It follows that the sum of the surfaceintegrals over the two ends is zero. Now the surfaceintegral over either end is half the circulation round its boundary; but since the lines representing pieces of area are to be drawn outwards in both cases, these boundaries must be gone round in opposite directions. Since then, when they are traversed in opposite directions the circulations are equal and opposite in sign, it follows that when they are traversed in the same direction the circulations are the same. Or, the circulation is the same round any two sections of a tube of spin.

When the tube is small, the spin at any part of it is inversely proportional to the area of normal section. For then the surface-integral over the section is approximately equal to the spin at any point of it multiplied by the area of the section; and we have seen that this surface-integral is constant. Hence a vortex-filament rotates faster in proportion as it gets thinner.

This shews us also that a vortex-filament cannot come to an end within the fluid, but must either return into itself, each vortex-line forming a closed curve, as in the case of a smoke-ring, or else end at the surface of the fluid, where the velocity no longer changes continuously; and consequently our previous reasoning does not apply. Such a vortex-filament may be formed by drawing the bowl of a teaspoon, half immersed, across the surface of a cup of tea; the filament goes round the edge of the submerged half of the bowl, and the two ends of it may be seen rotating as eddies on the surface.

VELOCITY-POTENTIAL.

203

IRROTATIONAL MOTION.

q

If it is possible to cover a contour by a cap such that there is no spin at any point of it, the circulation round the contour will be zero, since it is equal to twice the surface-integral of the spin, taken over the сар. Let Ρ and q be two points on such a contour paqb, then the circulation from p to q is the same along paq as along pbq. For

(paq) + (qbp)=0, or (paq) = (pbq). Therefore

Of two paths going from p to q, if it is possible to move one into coincidence with the other without crossing any vortex-line, the circulation along them is the same.

Where there is no spin, the motion is called irrotational. If there is no spin anywhere, so that the motion is irrotational throughout all space, the circulation from one point to another is independent of the path along which it is reckoned. Let a point o be taken arbitrarily, then for every point p in the body there is a certain definite quantity, namely, the circulation along any path from o to p. This is called the velocity-potential at p. If p be moved about so as to keep its velocity-potential constant, it will trace out a surface which is called an equipotential surface. It is clear that we may draw an equipotential surface through every point of space, and in this way we shall have a system of equipotential surfaces. There is no circulation along any line drawn on an equipotential surface; because the circulation from one point to another is equal to the difference of their velocity-potentials. (Circulation from p to q = circ. from o to q-circ. from o to p.)

Suppose, for example, that a body has a motion of translation. Then a plane perpendicular to the direction of motion will be an equipotential surface; for there is no component of velocity along any line in such a plane, and therefore the circulation along that line is zero. If we

choose any point in this plane for the point o, the velocitypotential for all points in the plane will be zero; and for all other points will be proportional to the distance from this plane, being positive on the side towards which the body is moving, and negative on the other side.

EQUIPOTENTIAL SURFACES.

In general, the equipotential surfaces are perpendicular to the lines of flow. We have already seen that if we suppose the velocity of every point of the body to be marked down at that point, so as to constitute a permanent diagram of the state of motion of the body at a given instant, then the rate of change of the circulation from o to p, when p moves in the diagram with unit velocity, is the component along the tangent to the path of p of the instantaneous velocity at the point p. Hence if we now use P to denote the velocity-potential at p, viz. the circulation from o to p, we shall have a P = v cos 0, where v is the magnitude of the instantaneous velocity at p, and the angle it makes with the direction of s. those directions which lie in the equipotential surface through p are such that there is no change of potential when p moves along them, or a,P=0. Hence either v= = 0 or cos 00; that is, if there is any velocity, it is perpendicular to the equipotential surface.

Now

If the motion of p is along a line of flow, cos = 1, and P=v; that is to say, the velocity at any point is the rate of change of potential per unit of length along a line of flow. Hence if we take two equipotential surfaces very near to one another, the velocity at various points on one of these surfaces will be inversely proportional to the distance between them, with an approximation which is closer the nearer the surfaces are taken to one another. For the difference of velocity-potential between a point on one and a point on the other is constant; and the rate of change of P per unit of length is inversely proportional to the distance required to produce a given change in P.

MANY-VALUED POTENTIAL.

205

Hence if we draw surfaces corresponding to the values 0. 1, 2,... of the velocity-potential, this system of surfaces will constitute a sort of diagram of the state of motion of the body. The velocity is everywhere at right angles to the equipotential surfaces, and where these are close together the velocity is large, where they are far apart it is small.

MOTION PARTLY IRROTATIONAL.

Suppose that in a mass of fluid there is a single vortexring of any form (i.e. a vortex-filament returning into itself), but that there is no rotation in any other part of the fluid. Consider a closed curve which is once linked with the ring, such as abc. The circulation round such a curve is equal to the circulation round a section of the vortex-filament, which we know to be the same for all sections; for the curve can be moved until it coincides with the section without crossing any vortex-line. Let the circulation round abc be called C.

We will now consider the circulation from a point o to a point p. Let the circulation along a path which goes from o top entirely outside the vortex-ring be called (op). A path like oxp, which goes through the ring, can be altered, without crossing any vortex-line, into the form orsr'p, in which it is made up of a path orr'p outside the ring, and a path rsr'

linked with the ring. Hence the circulation along oxp is made up of the circulations along these two paths, or it is (op)+ C. A path such as oyp, which is twice linked with the ring, may be altered into a path going outside the ring together with two such closed paths as rsr', and consequently the circulation along

y