29. Consider now the actual vortex made up of an infinite number of infinitely small vortex filaments. If these be of volumes inversely proportional to their vortex densities (§ 25), so that their circulations are equal, we now see from the constancy of the impulse that the sum of the resultant areas of all the vortex filaments remains constant; and so does the sum of their rotational moments: and the resultant areal axis of them all regarded as one system is a fixed line in space. Hence, as in the case of a vortex filament, the translation, if any, through space is on the average along its resultant axis. All this, of course, is on the supposition that there is no other vortex, and no solid immersed in the liquid, and no bounding surface of the liquid near enough to produce any sensible influence on the given vortex.

2. Experiments illustrating Rigidity produced by Centrifugal Force. By John Aitken, Esq.

If an endless chain is hung over a pulley and the pulley driven at a great velocity, it is well known that the motion so communicated to the chain has almost no tendency to change the form of the curve in which the chain hangs, and that the principal effect of the motion is to confer on the chain a quasi-rigidity which enables it to resist any force tending to alter its curvature.

This is only true in a general sense, and possibly may be true of some ideal form of chain; but in all chains we can experiment on there are forces in action in the moving chain which tend to cause the chain to depart from the form which it has while at rest.

I shall refer to these disturbing forces later on. As the disturbing forces in most chains are very small, we shall neglect them, and for the present suppose the centrifugal force just balances the tension at all points. The following experiments were made to illustrate the balance of these forces, to show that into whatever curves we may bend the chain when in motion, the centrifugal force has no tendency to alter these curves: that all forms are forms of stability, as far as the centrifugal force is concerned.

The first experiments were to show the effect of destroying the balance between the tension and the centrifugal force. In these experiments the links on the descending side of the loop were

allowed to fall on a platform, so that part of the chain lay loose on the platform, thus destroyed the tension produced by the centrifugal force at the lower part of the chain. The chain was made to take the same velocity as the driving pulley, by being pressed into contact with it by means of an elastic wheel.

I. When the chain was pressed at the point where it leaves the pulley there was no alteration in the path of the chain, because the chain after it leaves the pulley is moving in a straight line, and as there is no deviating force, there is no centrifugal force, and therefore, removing the tension in the chain has no effect on the direction of the motion of the links.

II. When the chain was pressed at a point a little higher up the pulley, then the centrifugal force of the curved part of the chain resting on the pulley at the descending side, being unbalanced by the tension, rises from the pulley, and is shot in a direction away to one side of the pulley. Of course the curved part of the chain on the other side of the pulley has also a tendency to rise, but is kept in its place by the tension produced by putting the chain in motion after being stopped by the platform.

III. When the chain is pressed on the ascending side of the pulley, then the chain rises up off the pulley and forms itself into a somewhat irregular curve resting on the platform, and touching the pulley at only one point. When the velocity is sufficient to raise it to a certain height, the conditions become altered. The chain in rising takes up all the slack chain lying on the platform, and a tension is produced in the chain by the centrifugal force, and unless we keep increasing the speed of the chain, it can no longer keep in its elevated position, because the centrifugal force is now balanced by the tension, and as the force of gravitation is now unbalanced, it gradually flattens the curve till the chain again comes to rest on the top of the pulley and spreads itself out in an irregular curve on the platform.

IV. At the beginning of the previous experiment the centrifugal force being unbalanced by the tension, it overcomes the force of gravitation and causes the chain to rise into the air. After all the slack chain has been taken up, and a tension is produced in the chain by the centrifugal force, then the centrifugal force is balanced by the tension and is no longer capable of opposing gravitation, and

the chain begins to fall; but at this point its fall may be stopped, or the chain may be made to rise again by destroying the tension. at the lower part of the chain. If we cause the chain, instead of meeting the platform at an acute angle, to strike it as near as possible at right angles, then the motion of the chain where it strikes the platform is partly destroyed, and the chain again rises and may be kept balanced for a long time resting on the platform, and only touching the driving wheel at one point. The reason for this being, that if we partially stop the motion of the links by causing them to strike the platform, or if we alter the direction of their motion by causing them to strike the platform, then there will be less tension in the lower part of the chain than in the upper, as the tension in the lower part will be only that due to partially changing the direction of the motion of the links. The centrifugal force of the upper part of the chain will be therefore unbalanced, and will cause the chain to rise and keep its elevated position against the force of gravitation. If a quick upward motion is given to the platform, the chain may be thrown up in the air, and again dropped on the platform like a solid body.

The next experiments are to show that centrifugal force may produce sufficient rigidity to cause a chain to run along a platform like a wheel. A short endless chain was put over a pulley which was driven at a great velocity; the chain was then dropped on the platform, along which it ran for some distance. It is not necessary that the chains form circular loops to do this. The loop may be tall and narrow, and will, while running along, keep the longer axis of the curve in its original upright position. Nor need the chain be heavy. A watch-guard was hung over a pulley about eight inches diameter; it then formed a loop about eight inches broad by about two feet high. When thrown off the pulley it glided along the platform for some distance. The chains were also dropped on an inclined polished surface, on which they remained standing in rapid motion for some time.

All these experiments only illustrate the balance of the centrifugal force, and the tension when the motion is all in one plane. The next experiment is to illustrate this balance when the motion takes place in different planes. This is easily illustrated by means of a circular disc of paper, or any other flexible material. If we

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bend the disc while it is rotating, we find that the bent part does not rotate with the disc, and that the disc only slowly regains its original flat form. If we load the outside of the disc with a row of flattened pellets of shot, we increase the resistance or rigidity of the disc while in motion, and if the weight is such that it just balances the elasticity of the paper, then the bend will remain in the same place for a very long time while the disc is rotating rapidly. The disc may even be bent till the circumference touches the centre, and while the bend keeps its place the chain of shot is passing through many planes, and the tension at the different points just balances the centrifugal force.

Before proceeding to experiment with the horizontal chain, I must refer to the disturbing forces in action causing the chain to change its form while in motion. When looking at these endless chains in motion, the most marked effect of this motion is to cause a curious reverse curve just after the chain has turned at the lowest point of its path and has begun to ascend. This reverse curve was supposed to be produced by friction from the great tension produced by the centrifugal force; but that it is not really so, is easily proved by taking two precisely similar chains and oiling one and passing the other through a flame to remove all grease. The only

difference between the two chains now is that the friction in the one is greater than in the other. If we hang these two chains over two pulleys of the same diameter on the same shaft so as to drive both chains at the same velocity, we find that the oiled chain has the reverse curve well marked, while the friction in the other chain causes the loop to open out and take up a curve approaching a circle and shows no reverse curve, and when both chains are compelled to have the same curvature at bottom, the reverse curve is much the least where the friction is greatest. The reverse curve seems to be due to the change of motion which takes place in the links when moving in a path of varying curvature. For instance, when the links are descending along the flat part of the curve, their motion is almost simply one of translation, whereas when passing round the curves they have a motion of rotation as well as a motion of translation, the result of which is, that the links resist this rotation at the entrance of the curve, and thus flatten out the curve on that side, and after the rotation has been communicated to them, they tend to

keep this rotation, and thus continue the curve at the lower end of the chain much farther round than if the chain was not in motion. And, for very evident reasons, the quickest part of the curve is not at the bottom, but a short distance up the ascending side; and farther, the rotation of the link at the bottom is quicker than that corresponding to the curvature. These points may all be illustrated by a chain in which the links are short and the chain as thick as possible, so that the moment of inertia of the links round an axis perpendicular to the plane of the motion of the links is as great as possible. Such a chain when properly made gives a series of large and well-marked waves all the way up the one side of the loop and down the other. The length of the links also tends to change the form of the curve. If we have two chains of the same material and same size every way, except in the length of the link, then the larger the link the more the chain tends to open out the curves and take the circular form, and the smaller the links the nearer it approaches the form it has while at rest, and the more marked the reverse curve becomes.

An elastic band in rapid motion will also tend to take up a circular form, because the strain at the quick part of the curves will tend to open them out, in the same manner as when the band was at rest. An elastic chain while in motion does not show the reverse curve like a chain, probably because the strain on the material prevents it doing so.

In the previous experiments gravitation acted on the chains, so that whatever form we might impress on them, gravitation constantly tended to change that form and bring it back nearly to the form it would have if gravitation alone acted on it. An attempt was therefore made to get quit of the disturbing effect of gravitation. Different ways were tried of effecting this, but none of them were thoroughly successful. The next experiment shows the most successful method tried, namely, suspension. The chain is hung by means of a number of fine cords to a circular disc, capable of rotating about a vertical axis placed as far above the chain as possible. The chain is driven by means of a rapidly revolving horizontal pulley running on a vertical axis, and to give sufficient friction the chain is pressed to the pulley by means of an elastic wheel. The centre of suspension is so arranged that it can be