Lord Kelvin: An account of his scientific life and work

by Andrew Gray

Fig. 14.

But the most wonderful of these mechanical arrangements is the machine for analysing the curves drawn by a self-registering tide gauge, so as to exhibit the constants of the harmonic curves, and thus enable the prediction of tidal heights to be carried out either by the tide-predicting machine, or by calculation. One day in 1876, Thomson remarked to his brother, James Thomson, then Professor of Engineering at Glasgow, that all he required for the construction of a tidal analyser was a form of integrating machine more satisfactory for his purpose than the usual type of integrator employed by surveyors and naval architects. James Thomson at once replied that he had invented, a long time before, what he called a disk-globe-cylinder-integrator. This consisted of a brass disk, with its plane inclined to the horizontal, which could be turned about its axis by a wheel gearing in teeth on the edge of the disk, and driven by the operator in a manner which will presently appear. Parallel and close to the disk, but not touching it, was placed a horizontal cylinder of brass, about 2 inches in diameter (called the registering cylinder), and between the disk and this cylinder was laid a metal ball about 2½ inches in diameter. When the disk was kept at rest, and the ball was rolled along between the cylinder and disk, the trace of its rolling on the latter was a straight horizontal line passing through the centre. Supposing then that the point of contact of the ball with the disk was on one side, at a distance from the centre, and that the disk was then turned, the ball was by the friction between it and the disk made to roll, and so to turn the cylinder. The angular velocity of rolling, and therefore the angular velocity of the cylinder, was proportional to the speed of the part of the disk in contact with it, that is, to y. It was also proportional to the speed of turning of the disk.

The mode by which this machine effects an integration will now be evident. Imagine the area to be found to lie between a curve and a straight datum line, drawn on a band of paper. This is stretched on a large cylinder, with the datum line round the cylinder. We call this the paper-cylinder. The distances of the different points of the curve from the datum line are values of y. A horizontal bar parallel to the cylinder carries a fork at one end and a projecting style at the other. The globe just fits between the prongs of the fork, and when the bar is moved in the direction of its length carries the ball along the disk and cylinder. When the style at the other end is on the datum line, the centre of the ball is at the centre of the disk, and the turning of the disk does not turn the cylinder. When the bar is displaced in the line of its own length to bring the style from the datum line to a point on the curve, the ball is displaced a distance y, and there is a corresponding turning of the cylinder by the action of the ball. In the use of the instrument the paper-cylinder is turned by the operator while the style is kept on the curve, and the disk is turned by the gearing already referred to, which is driven by a shaft geared with that of the paper-cylinder. Thus the displacement of the ball is always y, the ordinate of the curve, and for any displacement dx along the datum line, the registering cylinder is turned through an angle proportional to ydx. Thus any finite angle turned through is proportional to the integral of ydx for the corresponding part of the curve: a scale round one end of the registering cylinder gives that angle. Thomson immediately perceived that this extremely ingenious integrating machine was just what he required for his purpose. The curve of tidal heights drawn (on a reduced scale, of course) by a tide-gauge, is really the resultant of a large number of simple curves, represented by a series of harmonic terms, the coefficients of which are certain integrals. The problem is the evaluation of these integrals; and the method usually employed is to obtain them by measurement of ordinates of the curve and an elaborate process of calculation. But one of them is simply the integral area between the curve and the datum line corresponding to the mean water level, and the others are the integrals of quantities of the type y sin nx . dx, where y is the ordinate of the curve, and n a number inversely proportional to the period of the tidal constituent represented by the term.

All that was necessary, in order to give the integral of a term y sin nx . dx, was to make the disk oscillate about its axis as the paper-cylinder was turned through an angle proportional to x. Thus one disk, globe, and cylinder was arranged exactly as has been described for the integral of ydx, and with this as many others as there were harmonic terms to be evaluated from the curve were combined as follows. The disks were placed all in one plane with their centres all on one horizontal line, and the cylinders with their axes also in line, and a single sliding bar, with a fork for each globe, gave in each case the displacement y from the centre of the disk.

The requisite different speeds of oscillation were given to the disks by shafts geared with the paper-cylinder, by trains of wheels cut with the proper number of teeth for the speed required.

Thus the angles turned through by the registering cylinders when a curve on the paper-cylinder was passed under the style were proportional to the integrals required, and it was only necessary to calibrate the graduation of the scales of these cylinders by means of known curves to obtain the integrals in proper units.

One of these machines, which analyses four harmonic constituents, is in the Natural Philosophy Department at Glasgow; a much larger machine, to analyse a tidal curve containing five pairs of harmonic terms, or eleven constituents in all, was made for the British Association Committee on Tidal Observations, and is probably now in the South Kensington Museum.

But still more remarkable applications which Thomson made of his brother's integrating machine were to the mechanical integration of linear differential equations, with variable coefficients, to the integration of the general linear differential equation of any order, and, finally, to the integration of any differential equation of any order.

These applications were all made in a few days, almost in a few hours, after James Thomson first described the elementary machine, and papers containing descriptions of the combinations required were at once dictated by Thomson to his secretary, and despatched for publication. Very possibly he had thought out the applications to some extent before; but it is unlikely that he had done so in detail. But, even if it were so, the connection of a series of machines by the single controlling bar, and the production of the oscillations of the disks, all controlled, as they were, by the motion of a simple point along the curve, so as to give the required Fourier coefficients, were almost instantaneous, and afford an example of invention amounting to inspiration.

There should be noticed here also the geometrical slide for use in safety-valves, cathetometers and other instruments, and the hole-slot-and-plane mode of so supporting an instrument now used in all laboratories. These were Thomson's inventions, and their importance is insisted on in the Natural Philosophy.