up into a point without passing out of the region. Whenever this is the case, it does not follow from there being no spin within the region, that the circulation round such a curve is zero; for it may, as in this case, embrace a vortex line lying outside of the region.

VORTICES.

We shall next investigate the motion in which there is no spin except at a certain closed curve, and in which the velocity-potential is proportional to the solid angle subtended by this closed curve

at any point. By this we mean that from a point p lines are to be drawn to all points of the contour, forming a cone, and that this cone is to be cut by a sphere having its centre at p. The area which the cone marks off on the sphere, divided by the radius of the sphere, is the solid angle subtended at

p by the contour. Then the velocity-potential at Ρ is ΖΩ

where v is constant.

If we move the point p round the path pqr, the solid angle will diminish until it vanishes at some position near q. If we suppose a straight line passing through p to generate the cone, by moving round the contour in a definite direction, indicated by the arrows, the area on the spherical surface will be gone round in a definite way, by the intersection of the sphere with this moving line. We must then suppose the area on the left of the tracing point to be positive, and that on the right negative, p. 8. After the solid angle has acquired the value zero at q, it will change sign; and if we move our point on to r, the spherical area inside the cone must be reckoned negative. If we move on from r to p, passing through the contour, the area inside the cone at will change continuously into the area outside the cone at p; and this is to be reckoned negative. Hence by going round a closed

POTENTIAL OF SMALL VORTEX.

217

path which embraces the contour, we have continuously changed the solid angle into 2-4π. Hence the velocity-potential has by the same closed path been diminished by 4πv, because it is changed from v to v (-4π). It follows that the circulation round any path which embraces the contour is 4πv, if the path go round in the direction rqpr.

If therefore we consider a piece of the contour so short as to be approximately straight, the motion in its immediate neighbourhood will be like that round the axis of a whirl for which λ= 2v. As in that case, we may draw a small tubular surface enclosing the contour, and substitute for the actual motion inside of it that of a small vortex-filament; so that any small length of, this filament rotates like a cylinder about its axis. In this way we may make the velocity vary continuously, yet so that the motion is everywhere conceivable.

If we suppose the contour to be covered by a cap, and that the area of this cap is divided into a number of small areas, then the solid angle subtended by the contour at any point is the sum of the solid angles subtended at that point by all the small areas. Consequently the velocitysystem just described, which may be called a vortex, is the resultant of a number of smaller vortices, whose vortexlines are small closed curves which may be regarded as approximately plane. We shall now, therefore, examine more closely the case of such a small plane closed curve.

Take a point a within the area, and draw ax perpendicular to its plane. Let the angle xap=0, and the magnitude of the area = A. If we draw a sphere with centre p and radius pa, the area marked off on it by a cone with vertex p standing on A will be A cos nearly. For if the area is small, the portion of the sphere cut by the cone may be regarded as approximately plane, and the generating lines of the cone are approximately parallel,

8

so that the spherical area is very nearly an orthogonal projection of A. Hence the solid angle subtended at p is

nearly equal to A cose, if rap. Consequently the potential at p is vA cos: r2 approximately.

Now we can produce the same potential in another way. Let us put at b a source of strength μ, and at a a sink of the same strength; then the potential at p due to this combination is

μ μ

bp ap

Hence if we make vAμ. bc, and then let the area A and the length be diminish continually, increasing v and μ so as to keep vA, μ. bc, = k, a finite quantity, both the vortex and the combination of a positive and negative squirt will continually approximate to the motion in which the velocity-potential is k cos: r2. Now the source-and-sink combination gives no expansion except at a and b; consequently the limiting motion gives no expansion except But we have seen that every vortex may be made up of component-vortices, whose vortex-rings are as small as we like. Hence these two conclusions:

at a.

A.

There is no expansion anywhere due to a vortex.

2. A vortex is equivalent to a system of squirts constructed in this way. Let two caps be drawn covering the vortex-ring, so as to be everywhere at a very small distance from each other. Let one of them be continuously covered with sources and the other with sinks, so that the source and sink on any normal are equal in strength, and so also that if μSA be the total strength of the sources on the small piece of area SA, and t the thickness of the shell at that part, the product ut is constant all over the shell and equal to the constant of the vortex. Then, keeping all these conditions satisfied, this system of squirts will the more nearly approximate to the vortex the more nearly we make the two caps approach one another. For if we divide the cap into small areas, we have already seen that this is true of all the vortices whose vortex-rings are the boundaries of those areas.

Such a system of squirts is called a double shell. Inside the shell itself the velocity is not that due to the

POTENTIAL OF EXPANSION.

219

vortex, but is very large and in the contrary direction, namely, from the source to the sink. In crossing the shell the velocity potential is changed by 4πv.

VELOCITY IN TERMS OF EXPANSION AND SPIN.

We are now able to resolve the velocity-system of an infinite mass of fluid, having no velocity-potential at infinity, into squirts and vortices.

a

Let E be the expansion at a point a. Suppose the entire volume divided into small portions, of which 8Va is the one including the point a. Place at the point a a source whose strength is ESV : 4π. Then the rate of increase of 8V, due to this source, is ESV. And the velocity-potential at a point p, due to this source, is - ESV 4πap, where rap means the distance between a and p.

ар

If a similar source be placed at some point inside each of the small pieces SV into which the volume is divided, the velocity-potential due to all of them will be And if we indefinitely diminish the size and increase the number of the SV, this quantity will ap

ESV

Απγ

4πTup

[Ead Va

proximate to the integral - Eav..

The meaning of the integral, however, requires examination. It supposes that every point where there is expansion is a source, so that in a region where the expansion is constant, the sources will be uniformly distributed. The strength of the source at each point must be zero, since the aggregate strength of all the sources in a portion of the volume is the rate of increase of that portion of the volume, which is finite. We have therefore to form the conception of a continuous distribution of source over a volume, so that the aggregate strength is a finite quantity, and yet there is a source at every included point. If, for example, sources are uniformly distributed in the interior of a sphere, the effect will be a homogeneous strain-flux, consisting of a uniform expansion of the sphere;

so that the velocity relative to the centre is proportional to the distance from it. When the distribution is variable, the rate of distribution at any point is what would be the aggregate strength of a unit of volume in which the distribution was uniformly what it actually is at that point. If S is the rate of source-distribution at any point, E the expansion at that point, then E-4TS. For the rate of increase of a sphere of unit volume is 47 × the aggregate strength of the sources within it.

=

We have given, then, a velocity-system in which the expansion at any point a is E, and the velocity-potential is zero at an infinite distance. We construct the system

Απη ap

whose velocity-potential at a point p is -Ed; and we shew that this system also has expansion E at every point a (for only the sources in the immediate neighbourhood of a point can produce expansion at the point), and its velocity-potential is obviously zero at an infinite distance.

If therefore in the given system there is no spin, the given and the constructed systems are identical; for if we subtract one from the other, we get a system in which there is no spin, no expansion (for the expansions in the two systems are everywhere the same), and no velocitypotential at infinity. And this, we have already proved, means no motion at all.

at

Next, let there be spin in the given system, and let , be the solid angle which a certain vortex-line subtends p. Let a very small curve of area SA be drawn embracing the vortex-line, and let w be the spin at that part; then 2wdA will be the circulation round this curve. Let 2wdA = dk; then a motion whose velocity-potential at Ωδή

p is

will have no expansion anywhere and no spin except at the vortex-ring. If we suppose the spin to be confined to isolated vortex-filaments, we may draw a surface across each filament, divide this surface into small areas SA, and draw a vortex-ring through some point in each one of these small areas. The sum Σ Ωρδί will

4π