We may conveniently write (ab) for the circulation along ab. Thus (abcd) = (abca)+(acda).

In a homogeneous strain-flux, the circulation round two closed curves is the same if one can be made to coincide with the other by a step of translation. For if the positions of two corresponding points differ by the constant vector a, then the velocities differ by the constant quantity da; and the difference of the circulations is merely the length of pa multiplied by the projection of the closed curve on a line parallel to pa, which is of course zero.

Since the circulation round a closed curve is thus unaltered by the same velocity being given to all its points, we may if we like reduce any one point to rest, without altering the circulation round any closed curve.

a

с

f

d

The circulation round any two parallelograms of the same area is the same. We may change abdc into abfe by adding ace and subtracting bdf; and the circulation round these two triangles is the same. By repeating this process we may make one parallelogram into a translation of any other of equal area. By equal area is of course implied that they are in the same or parallel planes. The circulation round any parallelogram is double of that round a triangle of half its area. Let o, the middle point of ad, be brought to rest. Then the circulation along ad is zero, and the velocities at corresponding points of ab and dc being equal and opposite, the circulation along ab is equal to that along dc; similarly that along bd is equal to that along ca. Thus (ab) + (bd) + (da) = (ad) + (dc) + (ca), or the circulations round the triangles abd, adc are equal, and therefore each half of the circulation round abdc.

α

с

d

It follows that the circulation round any two triangles of the same area is the same.

Hence the circulations round any two areas in the same or parallel planes are proportional to those areas.

CIRCULATION IN TERMS OF SPIN.

197

For we may replace each of them by a polygon with short rectilineal sides, and these polygons may then be divided into small equal triangles. The areas will be nearly as the numbers of these triangles with an approximation which can be made as close as we like by making the triangles small enough. But the circulation round each polygon is the sum of the circulations round its component triangles; therefore the two circulations are also as the numbers of the triangles approximately, and therefore as the two areas exactly.

C

d

If a, B are the sides of a parallelogram, the circulation round it is SBpa - Sapß. Let a = ab, Bac. Then the sum of the circulations along ab and dc is the difference of those along ab and cd; which is the length of a multiplied by the resolved part of 8 along it, or -Szoß. Similarly the sum of the circulations along bd and ca is the difference of those along bd and ac, which is seen in the same way to be Sßpz. Hence the proposition.

a

a

The circulation round any plane area is equal to twice the product of the area by the component of spin perpendicular to it. A unit area in the plane oXY is the square whose sides are i,j. Now pi = ai +hj+g'k, øj=h'i+bj+fk; therefore Sioj - Sjpi = h — h', which is twice the component of spin round oZ. Now any plane whatever may be taken for the plane of oXY; whence the proposition.

STRAIN-FLUX NOT HOMOGENEOUS.

In the case of a homogeneous strain-flux, if we take any point p of the body and draw a straight line pq through it, the velocities of points on this line, relative to p, will all be parallel and proportional to the distance from p along the line. Consequently the rate of change of the velocity, as we go along the line pq, is constant.

When the strain-flux is not homogeneous, this rate of change of the velocity will no longer in general be con

stant.

But we may imagine a homogeneous strain-flux

which is such that the rate of change of velocity due to it, in any direction, is the same as the rate of change at p when we are moving in that direction in the actual condition of the body. This homogeneous strain-flux will then be called the strain-flux at p. It will in general vary from one point of the body to another.

In order that there may be a strain-flux at p at all, it is necessary that the velocity should change gradually as we pass through p in any direction. That is to say, there must be a rate of change up to p, and a rate of change on from p, and these must be equal. When this is the case, the entire strain-flux of the body may be said to be elementally homogeneous, or homogeneous in its smallest parts. Any small portion of the body moves with an approximately homogeneous strain-flux, and the approximation may be made as close as we like by taking the portion small enough. But if one portion of the body is sliding over another portion with finite velocity, this is not the case. In crossing the common surface of the two portions, we should find a sudden jump in the velocity. Such discontinuities have to be separately considered.

Let now a be any vector drawn through the point p; and let do, as before, mean the change that would be produced in σ by passing from one end of a to the other, if the rate of change per unit length remained uniformly what it actually is at p. Then the strain-flux at p has a velocity-function & such that do=p2. If therefore σ = ui+vj+wk,

the matrix of p is

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Consequently the spin w is

{{(Ə ̧w—Ə ̧v) i + (Ə ̧μ − Əμw)j + (Ə ̧v — Ə ̧u)k}.

It follows from this formula that if two velocity-systems are compounded together, the spin at any point in the resultant motion is the resultant of the two spins in the component motions.

LINES AND TUBES OF FLOW.

199

LINES OF FLOW AND VORTEX-LINES.

At every instant a moving body (to fix the ideas, consider a mass of water) has a certain velocity-system, i.e. every point in the body has a certain velocity σ. A curve such that its tangent at every point is in the direction of the velocity of that point is called a line of flow. It is clear that a line of flow can be drawn through any point of the body, so that at every instant there is a system of lines of flow. If the body has a motion of translation, the lines of flow are straight lines in the direction of the translation. If it rotates about an axis, the lines of flow are circles round the axis. If fluid diverges in all directions from a point, the lines of flow are straight lines through that point.

It is important to distinguish a line of flow from the actual path of a particle of the body. A line of flow relates to the state of motion at a given instant, and in general the system of lines of flow changes as the motion goes on. Thus while the path of a particle touches at every instant the instantaneous line of flow which passes through the particle, it does not in general coincide with any line of flow. The particular case in which the system of lines of flow does not alter, and in which, therefore, each of them is actually the path of a stream of particles, is called steady motion. In that case, the lines of flow are called stream-lines.

Thus, if a rigid body move about a fixed point, we know that its velocity-system at every instant is that of a spin about some axis through the fixed point, and consequently the lines of flow are circles about that axis. But in general the axis changes as the motion goes on, and the path of a particle of the body is not any of these circles.

If we take a small closed curve, and draw lines of flow through all points on it, the tubular surface traced out by these lines is called a tube of flow. In the case of steady motion all tubes of flow are permanent, and the portion of the body which is inside such a tube does not come out of it.

In general, a body has also at every instant a certain spin-system; i.e. at every point of the body there is a certain spin w. In fact, if the strain-flux is elementally homogeneous, there is at every point a homogeneous strainflux which is the resultant of a pure strain-flux and a spin w.

A curve such that its tangent at every point is in the direction of the spin at that point is called a vortex-line. If we draw vortex-lines through all the points of a small closed curve, we shall form a tubular surface which may be called a tube of spin; the part of the body inside the tube is called a vortex-filament. In the cases of fluid motion which occur most often in practice, there is a finite number of vortex-filaments in different parts of the fluid, but the remaining parts have no spin.

CIRCULATION IN NON-HOMOGENEOUS STRAIN-FLUX.

If we consider any small area Sz, which may be taken to be approximately plane, the strain-flux in its neighbourhood is approximately homogeneous; and if w be the spin at a point inside of the area, the circulation round the area will be approximately equal to its magnitude multiplied by twice the component of spin perpendicular to it; that is, it will be approximately - 2Swd1, where dx is regarded as a vector representing the area, and therefore perpendicular to it. This approximation is closer, the smaller the area is taken.

e

Now let abcd be any closed contour, whether plane or not, and let us suppose it to be covered by a cap, as aec, so that the contour is the boundary of a certain area on the surface of this cap. If this area be divided into a great number of very small pieces, as f, each of these may be taken to be approximately plane. And the circulation round abcd will be the sum of the circulations round all the small pieces. Thus it will be approximately equal to - 2 Swda,

α