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shades for which have also been hitherto mainly imported from abroad. This particular kind of glass is of pressing importance in relation to coal-mining, and it is certainly humiliating to learn from the makers of these lamps that for glass of the required quality capable of complying with the Home Office tests, we have been so largely dependent upon foreign glass manufacturers.
With respect to the supplies of chemical reagents, the joint committee found it necessary to entrust to a special sub-committee the somewhat arduous task of compiling a list of all the commonly used reagents with indications of the standard of purity required and the tests necessary for ascertaining whether the required standard had been reached. The list has been published as a pamphlet, and has been sent to many firms and companies of manufacturing chemists with the view of ascertaining which reagents of their own manufacture they are prepared to supply. When the replies have been received the joint committee will know the requirements of the profession could be met by British manufacturers.
It is hoped that sooner or later laboratory supplies both of apparatus and materials will be entirely of British origin. The manufacturers are-in some cases at considerable cost-developing lines of industry which are of the nature of new departures for this country. It is the duty of consumers and users-in fact, of every branch of the profession-to do their utmost to encourage and support these new home industries. Patriotism and the credit of our country alike demand that, after the war, they should help those who are helping them by insisting upon having nothing but the products of British manufacture. They should not only assist in the development of these industries now, but insure their permanent retention after the declaration of peace. With the achievement of this result there would be removed the reproach that the nation which gave to chemical science Priestley, Black, Boyle, Cavendish, Davy, Dalton, Faraday, and Graham-the country which founded the coal-tar colour industry, and which had taken the lead in the manufacture of "heavy chemicals," allowed her laboratory work to be dependent upon foreign materials, and her great textile and metallurgical industries to be threatened through the stoppage of supplies from inimical countries.
LORD KELVIN'S WORK ON GYROSTATICS.8
V.-Gyrostatic Theory of Elasticity. [Note.--In the explanation of steady precession, near the foot of the first column of p. 715 of NATURE of February 25, the words, "the horizontal axis A of the couple." referred to a cut, which, owing to an accident, could not be given. But Fig. 6 there printed, and repeated here on page 21. will serve instead. In that, as indicated in the small diagram at the bottom of the figure, the axis of angular momentum-the spin-axis-is to be supposed drawn towards the right, from the centre of the gyrostat along the (horizontal) axis of rotation, and the axis of the couple horizontally from the centre towards the observer. The dotted arc, marked 90°, should be continued round to the axis marked mkw. The angle 90° is that between the p.n-axis and the couple-axis.-A.G.]
NE other experiment I shall make with the veteran gyrostat, which has been spun again. You see that the rim carries two trunnions in line with the centre of the wheel (Fig. 10). These are placed on bearings attached to this square wooden frame; and now you see that as I hold the tray in my hands in a horizontal position, the gyrostat rests with its axis vertical or nearly so. The direction in which the wheel is spinning is shown by the arrow on the upper side. I now carry the tray round in azimuth in the direction of spin nothing happens; the gyrostat spins on 8 Abridged from the Sixth Kelvin Lecture, delivered at the Institution of Electrical Engineers, on January 28, by Prof. A. Gray, F. R.S. (Continued from NATURE, No. 2365, vol. xciv.,,p. 716.)
placidly. If, however, I carry the tray slowly round the other way, the gyrostat immediately turns upside down on the trunnions; and now, as I go on carrying the tray round in the same direction as before, the gyrostat is quiescent as at first; but the spin, by the inversion of the gyrostat, has been brought into the same direction as the azimuthal motion.
The gyrostat behaves as if it possessed volitiona very decided will of its own. It cannot bear to be carried round in the direction opposed to the rotation, and, as it cannot help the carrying round, it accommodates itself to circumstances by inverting itself so that the two turning motions are made to agree in direction. Again I reverse the azimuthal motion, and the gyrostat inverts itself so that the wheel turns in the same direction in space as at first.
The inversion brings into play a wrench on the hands of the experimenter. A varying couple, lasting during the time of the inversion, is required to reverse the angular momentum of the wheel in space, and this is applied to the gyrostat by the frame at the trunnions, and to the frame, because that is kept steady, by the hands of the operator. The total change of angular momentum is 2 N, where N is the angular momentum of the flywheel, and this is the time-integral of the couple.
It will be noticed that in this experiment, in which the gyrostat displays this curious one-sided stability and instability, it is affected by a precession impressed upon it from without. The system was not
left to itself, I carried it round. The gyrostat had little or no gravitational stability—the centre of gravity was nearly on a level with the trunnions; but even if it were gravitationally unstable, sufficiently rapid azimuthal motion would keep it upright if that motion agreed with the spin, while the least motion the other way round would cause it to capsize.
It is important to notice that if the gyrostat be placed on the trunnions, so that the axis of the wheel is in the plane of the frame, azimuthal turning in one direction causes one end of the axis to rise, or turning in the other direction causes the other end to rise. As I shall show presently, this means a reaction couple on the frame which must be balanced by a couple applied by the experimenter.
Better than anything else I know, this experiment of the capsizing of the gyrostat by azimuthal motion affords an example of the two forms of solution of a certain differential equation, which, when the gyrostat is without sensible gravitational stability, and is small, I may write
we have a solution giving oscillations about the vertical, in the period 2A/N: the equilibrium is stable. When, however, w is reversed the product must be given the opposite sign, and we get a solution in real exponentials, starting falling-away from the upright position, which is continued until the opposite (stable) position is attained. N has now also been reversed in space, and the product w N in the differential equation is again positive.
As I have already stated, the time integral of the turning motive about the vertical required by the gyrostat from the frame constraining it to move round in azimuth is 2 N; that is, 2 Cn, where C is the moment of inertia of the flywheel about its axis. There is thus at each instant of the turning in azimuth before the inversion has been completed a couple required from the frame, and this couple is greater the greater the angular speed n of spin.
The couple arises thus. Let the gyrostat axis have been displaced from the vertical through an angle @ about the trunnion axis. In consequence of the azimuthal motion, at rate w, say, the outer extremity of the axis of angular momentum is being moved parallel to the instantaneous position of the line of trunnions, and thus there is rate of production R of angular momentum about that line; but there being no applied couple about the trunnions, the gyrostat must begin to turn about the trunnions to neutralise R. This turning tends to erect or to capsize the gyrostat according as the spin and azimuthal motions agree or are opposed in direction. In its turn, however, this involves production of angular momentum about the vertical for which a couple must be applied by the frame, and of course to the frame by the operator. This couple is greater the greater Cn, and therefore if the operator cannot apply so great a couple, an azimuthal turning at rate w cannot take place. With sufficiently great angular momentum the resistance to azimuthal turning could be made for any stated values of 0 and greater than any specified amount.
The magnitude of this couple which measures the resistance to turning at a given rate is greatest when the angle is 90°; that is, when the axis of the flywheel is in the plane of the frame.
Now I come to an interesting application of these ideas. You are aware that Lord Kelvin endeavoured to frame something like a kinetic theory of elasticitythat is, he conceived the idea that, for example, the rigidity of bodies, their elasticity of shape, depends on motions of the parts of the bodies, hidden from our ordinary senses, as the flywheel of a gyrostat is hidden from our sight and touch by the case. Look at this diagram of a web (Fig. 11). It represents two sets of squares, one shown by full, the other by fine, lines; the former are supposed to be rigid squares, the latter flexible. Unlike ordinary fabrics, which are almost unstretchable except in a direction at 45° to the warp and woof, this web is equally stretchable in all directions. If the web is strained slightly by a small change of each flexible square into a rhombus, or into a not-square rectangle, the areas are to the first order of small quantities unaltered.
Now imagine that a gyrostat is mounted in each of the rigid squares, so that the axis of the trunnions and the axis of rotation are in the plane of the square as shown in Fig. 12. If the angular speeds of the flywheels are sufficiently great, it is impossible to turn the squares in azimuth at any given small angular speed. Thus any strain involving turning of the small squares is resisted, and we have azimuthal rígidity conferred on the web by the gyrostats. There
rotation, but now about any axis whatever; and there would be no resistance to mere translation of the cubes as wholes. Thus the body so constituted would be undistinguishable from an ordinary elastic solid as regards translatory motion, but would resist turning.
It is convenient in this connection to refer to an arrangement—a gyrostatic imitation of a spiral spring -in which a constant displacement is produced and maintained by the action of a constant force in a fixed direction, involving the application of a couple of constant moment, though not of constant direction of axis. This gyrostatic spring balance is indicated in a paper entitled "On a Gyrostatic Adynamic Constitution for Ether," published partly in the Comptes rendus," and partly in the Proceedings of the Royal Society of Edinburgh. 10 This is one of the many papers which Lord Kelvin published in the latter part of his life on a question that occupied him much from time to time, the nature of the æther as a vehicle of light and as the medium in which electric and magnetic phenomena are manifested.
The spring balance is described in some detail in his "Popular Lectures and Addresses."" I had thought of realising the arrangement, which is shown in Fig. 13, but on consideration I found that though it would act as a spring, it would not, except under certain conditions, not easily realisable even approximately, possess the peculiar property of a spiral spring of being drawn out a distance proportional to the 9 Comptes rendus, vol. cix., p. 453, 1889. Math. and Phys. Papers, vol. iii., p. 466.
10 Proceedings of the Royal Society or Edinburgh, vol. xi., 1890. 11 Vol. i., p. 237, et seq.
weight hung on the lower hook. The gyrostatic arrangement is very difficult to realise with ordinary gyrostats, but presents no difficulty with our motor instruments. You see what the arrangement is. A frame of four equal bars is constructed, by jointing the bars freely together at their extremities in the manner shown by the diagram. It is hung from a vertical swivelling pin at one corner, so that one diagonal of the frame is vertical, and another vertical swivelling pin at the lowest corner carries a hook. Four equal gyrostats are inserted, one in each bar, as shown, with its axis along the bar, and they have equal rotations in the directions shown by the circular arrows. Under the couples tending to change the directions of the axes of the flywheels, and applied by the weights of the gyrostats and bars, the system precesses round the two swivels, and so preserves a constant configuration. If now a weight is hung on the hook at the lower end, the frame is elongated a little, and a new precessional motion gives again a constant configuration of the frame, different, of course, from the former one. Two gyrostats, the upper or lower pair, would serve quite well to give the effect.
Lord Kelvin suggested that if the frame were surrounded by a case, leaving only the swivel-pins at top and bottom protruding, it would be impossible, apart from special knowledge of the construction of the interior, to discern the difference between the system and an enclosed spiral or coach spring, surrounded by a case and fitted with hooks for suspension and attachment of weights. But unless the masses of the gyrostats are very small (while their angular momenta are exceedingly great), so that the change of kinetic enegy due to the change in precessional motion may be put down entirely, or nearly so, to the work done by gravity on the weight carried by the hook, in its descent from one configuration of steady motion to another, the distance through which the frame is lengthened is not simply proportional to the load applied.
A fair idea of the action, and, indeed, an approximate realisation of the property aimed at, is obtained by means of the arrangement shown in Fig. 6 above. We have had it before. A gyrostat is hung with its axis horizontal by a cord in the same vertical as the centroid. The flywheel spins, but as there is no couple there is no precession. A weight mg is applied in a vertical line at distance from the centroid, as indicated by the diagram; a slight, very slight, tilting of the gyrostat is produced, and the gyrostat moves off with not quite steady precession, of average angular speed μ. Neglecting the slight deviation now set up of the suspension cord from the vertical, and putting A for the moment of inertia of the gyrostat about a vertical axis through its centre, we get for the kinetic energy of the azimuthal motion the value § A μ2 + †m l2 μ2. The work done by the weight mg in its descent through the small distance h involved in the tilting is mg h. Hence we get (A+m l2) μ2 = m gh.
As we have already seen, however, we have in this case μm gl/C n. Substituting in the equation just found this value for μ, and supposing that A is great in comparison with m 12, so that the term ml3μ3
Thus h is proportional to m.
It will be evident that if on the right-hand side of the first equation there had been terms due to descent of the gyrostat through a distance of h or h, this equation of proportionality could not have been obtained.
The idea, however, underlying the arrangement is very suggestive, and carries us a long way towards obtaining a definite notion as to how the elastic properties of bodies may be explained.
(A gyrostatic pendulum was here shown. See for figure and description NATURE, April 17, 1913.) VII.-Vibrations and Waves in Stretched Chain of Gyrostats.
[An account of this paper was given in the lecture, and will be found (with details of the mathematical discussion appended) in the Journal of the Institution of Electrical Engineers.]
VIII. Gyrostatic Observation of Rotation of the Earth.
The famous French experimentalist, Léon Foucault, suggested two ways of determining the rotation of the earth. One was observation of the apparent turning of the plane of vibration of a long pendulum, suspended so as to be as nearly as possible free from any constraint due to the attachment of the pendulum wire to its fixed support. This classical experiment was carried out with fair success at the Panthéon at Paris, and was repeated under the domes of the cathedrals of Amiens and Rheims. If the exponents of "Kultur" in Northern France were aware of this fact, they seem to have attached to it just as little weight as they gave to the more sacred
associations of the beautiful old church of the latter city.
Foucault's other method was based on the fact that a gyrostat, if mounted properly, retains unaltered the direction of the spin-axis when the supports are turned round. Here, for example, is our pedestal gyrostat, mounted freely in its enclosing frame which is carried by a vertical rod, swivelling in a vertical socket carried by the supporting stand (see Fig. 7 above). I can set the spinning gyrostat with its axis in any direction I please, and, when I turn the supporting stand round, a friction couple of some little magnitude is applied to the vertical rod. You see that I do not alter the direction of the spin-axis perceptibly. Yet the friction couple is sufficient to carry the gyrostat round with the stand when there is no spin. The spin results in a great increase of virtual inertia for turning displacements, as we shall see quantitatively in the case of one of Lord Kelvin's experiments, which I am about to describe.
In practice it is found desirable to subject the gyrostatic apparatus to a constraint which is perfectly definite; for example, the axis of spin may be kept horizontal. Solutions of the problem are to be found in the gyrostatic compasses now in use on the warships of various navies.
At the British Association meetings at Southport and Montreal, in 1883 and 1884, Lord Kelvin suggested methods of demonstrating the earth's rotation, and of constructing a gyrostatic compass. One of these had reference to the component of rotation about the vertical, the component, in fact, demonstrated by the Foucault pendulum experiment. If o be the resultant angular speed, the component about the vertical at any place in latitude l is sin l, while
the companion component about the horizontal there is cosi. Thus at London the component about the vertical is o'78 of w, and the period of rotation about the vertical is about 30'77 hours of sidereal time.
Lord Kelvin's method of measuring w sin l consists in supporting a gyrostat on knife-edges attached to the projecting edge of the case, so that the gyrostat without spin rests with the axis horizontal or nearly so. For this purpose the line of knife-edges is laid through the centre of the flywheel at right angles to the axis, and the plane of the knife-edges is therefore the plane of symmetry of the flywheel perpendicular to the axis. The knife-edges are a little above the centre of gravity of the instrument, which we suppose in or nearly in that plane, so that there is a little gravitational stability. The azimuth of the axis is a matter of indifference, as any couple due to the component of rotation about the horizontal is balanced by an equal couple furnished by the knife-edge bearings.
At points in a line at right angles to the line of knife-edges, and passing through it, two scale-pans are attached to the framework, and by weights in these the axis of the gyrostat (without spin) is adjusted, as nearly as may be, in a horizontal position which is marked. The gyrostat is now removed, to have its flywheel spun rapidly, and is then replaced. It is found that the weights in the scale-pans have to be altered now to bring the gyrostat back to the marked position. From the alteration in the weights the angular speed about the vertical can be calculated.
To fix the ideas, let the gyrostat axis be north and south, and let the spin to an observer, looking at it from beyond the north end, be in the counterclock, or positive direction. The rotation of the earth about the vertical carries the north end of the axis round towards the west, and therefore angular momentum is being produced about a horizontal axis drawn westward, at a rate equal to Cno sin l, where C is the angular momentum of the flywheel. If the sum of the increase of weight on one scale-pan and the diminution (if any) in the other be w, and a be the horizontal distance between the points of attachment of the scale-pans, we have
This gives w=0'047 gramme, or 47 milligrammes. It would require careful arrangements to carry out the experiment accurately, but the idea is clearly not unpractical. With some of the new gyrostats that we now have, the mass of the wheel is as much as 2000 grammes, and the radius of gyration is about 75 cm. These numbers bring w up, to 0.82 gramme, at the same speed.
If the gravitational stability of this gyrostatic balance be removed, that is, the line of knifeedges be made to pass accurately through the centre of gravity of the system of wheel and framework, and the axis of the wheel be placed in a truly north and south vertical plane, so that the knife-edges are horizontally east and west, the gyrostat will be in
stable equilibrium when the axis is parallel to the earth's axis, and is turned so that the direction of rotation agrees with the rotation of the earth. For we have then simply the experiment, described above, of the gyrostat mounted on trunnions resting on bearings attached to a tray which is carried round by the experimenter. The axis of the gyrostat was at right angles to the tray, and we saw that when the tray, held horizontally, was carried round in azimuth the equilibrium of the gyrostat was stable or unstable, according as the two turnings agreed or disagreed in direction. In the present case the tray is the earth, the position of the axis of rotation parallel to the earth's axis replaces the vertical position, and the earth's turning the azimuthal motion. If displaced from the stable position_the_gyrostat will oscillate about it in the period 2 A/C nw, where A is the moment of inertia about the knife-edges, and the other quantities have the meanings already assigned to them.
If the line of knife-edges be north and south, the vertical will be the stable, or unstable, direction of the axis of rotation, and there will be oscillation about the stable position in the period
2 A/C no sin l.
The gyrostat thus imitates exactly the behaviour of a dipping needle in the earth's magnetic field, and thus we have Lord Kelvin's gyrostatic model of the dipping needle.
It is right to point out that these arrangements were anticipated by Gilbert's barogyroscope, 12 which rests on precisely the same idea, and applies it in a similar manner.
At Montreal Lord Kelvin described a "gyrostatic model of a magnetic compass." This was one of his gyrostats hung, with its axis of rotation horizontal, by a long fine wire, attached to the framework at a point over the centre of gravity of the system, and held at the upper end by a torsion-head capable of being turned round the axis of the wire. By means of this torsion-head any swinging of the gyrostat in azimuth round the wire was to be checked until, when the head was left untouched, the gyrostat hung at rest.
In small azimuthal oscillations of the gyrostat about the axis of the wire, the wire being fixed to the gyrostat at the lower end and held by the head at the upper, the virtual moment of inertia of the gyrostat about the wire is greatly enhanced by the rotation of the flywheel If there were no rotation the moment of inertia would be A; with rotation it is virtually A(1 + C2n2/A M a g), where M is the whole suspended mass, a the distance of the point of attachment of the wire above the centre of mass gravity of the M. This will be found proved very simply in the Mathematical Appendix [see Journal, I.E.E.] to this lecture. It will be shown, moreover, that when the whole motion is considered-the tilting motion as well as the azimuthal-it appears that there are two fundamental periods of vibration. There is the long period due to the slight torsional rigidity of the long wire, and the enhanced moment of inertia pointed out by Lord Kelvin, and also a short period, the shortness of which is most properly to be reckoned as due to virtual diminution of the moment of inertia of the gyrostat, in the tilting motion, in exactly the same ratio as the other moment of inertia is increased. Both these periods are separately possible, and in
12" Mémoires sur divers problèmes," etc. Annales de la Société Scientifique, Bruxelles, 1877-8.
the most general motion they are superimposed. This second curious effect was not referred to by Lord Kelvin, and I have not seen it noted before. The co-existence of long and short periods is, however, characteristic of rapidly spinning gyrostatic systems.
Now suppose that the wire so long and of so slight torsional rigidity that this rigidity cannot stabilise the gyrostat in the position of unstable equilibrium. Then the effect of the component of rotation of the earth about the vertical is to produce tilting of the axis of the flywheel from the horizontal position, since this turning gives a rate of production of angular momentum about a horizontal axis at right angles to that of rotation. A slight tilt suffices to give an equilibrating couple, and so we can have the gyrostatic axis in a north and south vertical plane, and nearly horizontal, while the wire is without twist. Into this position the gyrostat is guided by manipulation of the torsion-head. The effect of the horizontal component of the earth's angular speed is now practically zero.
If now by the use of the torsion-head the gyrostat axis be brought to rest in a nearly horizontal position at an angle with the north and south horizontal direction, the component of turning about this position of the axis is cosl cos, and about a horizontal line at right angles to the new position of the axis is cos l sin . The former has no influence on the gyrostatic axis, the latter gives a rate of production of angular momentum about the vertical amounting to Cno cos I sin . Hence a couple of moment equal to this must be applied by means of the torsion-head to produce equilibrium, and this as we see is proportional to sin ø.
X.-General Dynamical Theory of Gyrostatic System with any Number of Freedoms.
I cannot do more than mention the gyrostatic investigations contained in the second edition of "Natural Philosophy." These were written while the proofs of $ 345 of the book were in his hands, and consist of additional sections (§§ 345-345xxviii) interpolated at that stage. From many points of view this part of the book is exceedingly interesting. It continues a subject which was also expanded in the same way on the proof-sheets (in §§ 343a-343m), that of oscillatory motion. Oscillatory motion for systems of two, three, four, six, or more freedoms with gyrostatic domination is enough to tax the skill of the most expert analyst, for questions arise regarding the roots of the determinantal equations and their interpretation, which require great care in handling. I may only quote the general conclusions as to gyrostatic domination.
Let the number of the freedoms be even (that is, the freedoms exclusive of those by which the flywheels have angular momenta about their axes). Let the equilibrium of the system when at rest (without spin of the flywheels) be either stable, or unstable, for every freedom. If the wheels are so linked up to the system as to render gyrostatic domination possible, then with sufficiently rapid spin the equilibrium becomes stable, with half the whole number, 2 n say, of its periods of vibration exceedingly small, and the other half very large. Each set of periods is given by the roots of a determinantal equation of degree n. The latter periods are to the first degree of approximation independent of the applied forces, and were called "adynamic," the former periods were called " 'precessional," and do depend on the applied forces.
The first approximations to the fast and slow azimuthal motion of a top are in point. The angular speed Cn/A cos does not depend on any applied forces, the other speed M gh/Cn does.
XI.-Difficulties of Mechanical Hypotheses and Models. Conclusion.
I have now dealt with most of Lord Kelvin's investigations and theories. These were related in many ways to electricity and magnetism, and in all of them he ever sought some dynamical explanation that would work. This, indeed, was the distinguishing feature of all his researches, the bringing of everything down to dynamics, and the construction where possible of illustrative mechanical models. Electricity and magnetism are highly dynamical affairs; we send signals by wire or wireless," we transmit power in a wonderful manner by an agency which we are still far from completely understanding, an agency which causes absorption of energy at one place on the earth's surface, and evolution again of a large portion of that energy at another place. The vehicle is the æther, for, in spite of all that I have been able to learn regarding the new theory of relativity, I still believe in the æther's existence. In all this we are held fast by dynamical laws, no doubt not yet formulated in full detail, but to a considerable extent already correctly comprehended.
Lord Kelvin certainly had confidence in his own theories and clung firmly to his conclusions. He was tenax propositi, yet he could on occasion acknowledge that he had made a mistake. His genius ranged over the whole field of physical science; no problem was too great or too small to attract his attention. No obstacles, no complications, daunted his spirit of inquiry. The thunders of Jove, the birth of the world and the cold death prepared for it by dissipation of energy, the harnessing of the energies of nature for the service of man, the guidance and safety of mariners, the genesis of waves and their breaking into spray and spindrift, all these questions, and many others, engaged his thoughts, to the lasting benefit of humanity and the increase of knowledge. Throughout all he was keen and calm and dispassionate, a truly unaggressive and kindly natural philosopher.
The function of science is to enable man to penetrate the secrets of nature, and to apply that knowledge to the promotion of the welfare and happiness of all living beings. No one would have repudiated with more scorn than Lord Kelvin that emanation of the Pit, the modern doctrine that culture—scientific, philosophical, or artistic-entitles a self-appraised and self-chosen nation to wade through seas of blood to the domination of the world.
UNIVERSITY AND EDUCATIONAL
LONDON. The report of the Military Education Committee to the Senate of the University for the year 1914 shows that very valuable service has been rendered to the country by the University Contingent of the Officers Training Corps. The number of cadets and ex-cadets of the Contingent gazetted to commissions between August 5, 1914, and the end of the year was 773, and the number of graduates and students of the University (not being past or present cadets of the O.T.C.) gazetted to commissions during the same period was 156. Allowing for officers commissioned from the O.T.C. before August 5, the total number of officers now serving, who are ex-cadets of the University of London O.T.C., or were recommended for their commissions by the University, is estimated at 1100. In addition, a large number of graduates and students of the University have been granted commissions through other channels or are