centre. If we take two of these very near to one another, the normal distance between them is everywhere the same; but we have shewn that the velocity, at different points of an equipotential surface, is inversely proportional to the normal distance of a contiguous surface. Hence it follows that the velocities are equal at equal distances from s.

The tubes of flow are cones having s for vertex. Let sac be such a cone, and let it be cut at ab and cd by spheres having their centres at s. The figure sab is similar to the figure scd; hence the areas ab and cd are to one another as saa to sc2.

a

d

The area ab divided by sa2 is called the solid angle of the cone at s. It is a spherical measure of the solid angle, just as the arc of a concentric circle, divided by the radius of that circle, is the circular measure of a plane angle.

Now the rate of increase of the volume abdc is equal to the surface-integral of the velocity over its boundary. This boundary consists partly of the side of the cone and partly of the spherical ends ab and cd. The side of the cone can contribute nothing to the surface-integral, because at every point of it the direction of the velocity is in the surface, and consequently there is no component normal to the surface. The spherical surface ab, being everywhere perpendicular to the velocity, which is constant all over it, supplies a portion of the surfaceintegral, which is simply the product of the area ab by the velocity at any point of it, say a. Similarly the surface-integral due to cd is the product of the area cd by the reversed velocity at c. But since there is no expansion, these two must be equal and opposite in sign; therefore velocity at a x area ab = velocity at cx area cd; or, which is the same thing,

velocity at a × sa2 = velocity at c × sc2.

Or, we may say that the rate at which the fluid flows across ab, is the area ab multiplied by the velocity at a, and that if there is no expansion, the fluid must flow across cd at the same rate.

We learn thus, that in the motion considered, the velocity is inversely as the square of the distance from s. Let v be the magnitude of the velocity at distance r, then vr2 is a constant, which we shall call μ. The circulation

along a straight line sab from a to b is then

and hence if we make the velocity-potential zero at an infinite distance its value at distance r will be

μ

The rate of increase of any sphere of radius r, having its centre at s, is equal to the velocity at any point of its surface multiplied by the whole surface of the sphere. Now the surface of a sphere is 47r2, and therefore the rate of increase is 4πrv, which is 4πμ. We should have expected this to be a constant, because there is no expansion in the space between two such spherical surfaces.

At the point s itself, the velocity-potential, the velocity, and the expansion, are all infinite, and we have no means of conceiving such a state of motion. To avoid this, we may imagine a very small sphere to be drawn round the point s, and the motion inside of this sphere to be replaced by a homogeneous strain-flux with the point s at rest, and the same velocity as in the original motion at all points on the surface of this sphere. The velocity will then vary continuously, and the motion will be conceivable at every point. Let E be the expansion of the homogeneous strainflux, V the volume of the small sphere, then EV=4πμ.

The point s is called a source of strength μ when the μ fluid streams out in all directions; when u is negative, so that the fluid streams inwards, it is called a sink. The whole velocity-system here described may be called a squirt.

WHIRLS.

Suppose next that the lines of flow are circles having their centres on a fixed axis, and their planes perpen

dicular to it, and that there is no spin except at the axis, and no expansion anywhere. Then the equipotential surfaces must be planes passing through the axis, and the velocity, being inversely proportional to the distance between two contiguous equipotential surfaces, must, for points on the same plane, be inversely proportional to the perpendicular distance from the axis. The condition that there shall be no expansion requires the velocity to be constant all round a circular line of flow. If the velocity at distance r from the axis be λr, where X is a constant, the circulation round any line of flow will be the length of it, 2πr, multiplied by the velocity λr; that is, it will be 27. The motion at the axis is inconceivable, as the velocity and the spin are infinite; but we may avoid this difficulty by drawing a very small cylinder round the axis, and supposing this to rotate about the axis as a rigid body, so that the points on its surface have the same velocity as in the original motion. This cylinder may then be regarded as an infinitely long straight vortexfilament, the circulation round any section of which is 2πλ.

If we suppose a region of space to be marked out by a surface like the surface of a ring, and the axis to pass

through the hole of this ring, but not into the region itself; then there will be no spin at any part of the region, and yet the fluid will flow continually round it. This explains how it is possible for fluid to flow continually round a re-entering channel, without ever having any motion of rotation. It is then possible to draw a closed curve within the region, which cannot be shrunk

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 where v is constant.

is ΖΩ

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 9: 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 r 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 47v, because it is changed from v to v (2–4π). 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,

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