The motion was defined as that of a particle moving along the diameter of a circle—the "auxiliary circle," Thomson called it—so as always to keep pace, as regards displacement in the direction along that diameter, with a particle moving with uniform speed in the circle. Then the velocity and acceleration were found, and it was shown that the particle was continually accelerated towards the centre in proportion to the distance of the particle from that point. The constant ratio of acceleration to displacement was proved to be equal to the square of the angular velocity in the auxiliary circle, and from this fact, and the particular value of the acceleration when the particle was at either end of its range of motion, an expression for the period in terms of the speed and radius of the auxiliary circle was deduced. Then the ordinary simple pendulum formula was obtained.
This mode of treatment of an elementary matter, so entirely different from anything in the ordinary text-books, arrested the attention at once, and conveyed, to some at least of those present, an idea of simple harmonic motion which was directly applicable to all kinds of cases, such as the motion of the air in a sound wave, or of the medium which conveys the waves of light.
The subject of Kepler's laws was dealt with in the early lectures of every course, and Newton's deductions were insisted on as containing the philosophy of the whole question, leading, as they did, to the single principle from which the laws could be deduced, and the third law corrected when the mass of the planet was comparable with that of the sun. Sometimes Thomson would read the remarkable passage in Hegel's Logik, in which he refers to the Newtonian theory of gravitation and says, "The planets are not pulled this way and that, they move along in their orbits like the blessed gods," and remark upon it. On one occasion his remark was, "Well, gentlemen, if these be his physics, what must his metaphysics be?" And certainly that a philosopher should deny, as Hegel seemed to do, all merit to the philosophical setting in which Newton placed the empirical results of Kepler, is a very remarkable phenomenon.
The vivacity and enthusiasm of the Professor at that time were very great. The animation of his countenance as he looked at a gyrostat spinning, standing on a knife-edge on the glass plate in front of him, and leaning over so that its centre of gravity was on one side of the point of support; the delight with which he showed that hurrying of the precessional motion caused the gyrostat to rise, and retarding the precessional motion caused the gyrostat to fall, so that the freedom to "precess" was the secret of its not falling; the immediate application of the study of the gyrostat to the explanation of the precession of the equinoxes, and illustration by a model of a terrestrial globe, arranged so that the centre should be a fixed point, while its axis—a material spike of brass—rolled round a horizontal circle, the centre of which represented the pole of the ecliptic, and the diameter of which subtended an angle at the centre of the globe of twice the obliquity of the ecliptic; the pleasure with which he pointed to the motion of the equinoctial points along a circle surrounding the globe on a level with its centre, and representing the plane of the ecliptic, and the smile with which he announced, when the axis had rolled once round the circle, that 26,000 years had elapsed—all these delighted his hearers, and made the lecture memorable.
Then the gyrostat, mounted with its axis vertical on trunnions on a level with the fly-wheel, and resting on a wooden frame carried about by the professor! The delight of the students with the quiescence of the gyrostat when the frame, gyrostat and all, was carried round in the direction of the spin of the fly-wheel, and its sudden turning upside down when the frame was carried round the other way, was extreme, and when he suggested that a gyrostat might be concealed on a tray of glasses carried by a waiter, their appreciation of what would happen was shown by laughter and a tumult of applause.
Some would have liked to follow the motions of spinning bodies a little more closely, and to have made out clearly why they behaved as they did. Apparently Thomson imagined the whole affair was self-evident, for he never gave more than the simple parallelogram diagram showing the composition, with the already existing angular momentum about the axis of the top, of that generated about another axis, in any short time, by the action of gravity.
As a matter of fact, the stability and instability of the gyrostat on the tray give the best possible illustration of the two different forms of solution of the differential equation, Ӫ + μӨ = 0, according as μ is positive or negative; though it is also possible to explain the inversion very simply from first principles. All this was no doubt regarded by Thomson as obvious; but it was far from being self-evident to even good students of the ordinary class, who, without exception, were beginning the study of dynamics.
Thomson's absorption in the work of the moment was often very great, and on these occasions he much disliked to be brought down to sublunary things by any slight mischance or inconvenience. Examples will occur to every old pupil of the great emphasis with which he commanded that precautions should be taken to prevent the like from happening again. Copies of Thomson and Tait's Natural Philosophy—"T and T'" was its familiar title—and of other books, including Barlow's Tables and other collections of numerical data, were always kept on the lecture-table. But occasionally a laboratory student would stray in after everything had been prepared for the morning lecture, and carry off Barlow to make some calculation, and of course forget to return it. Next morning some number would be wanted from Barlow in a hurry, and the book would be missing. Then Thomson would order that Barlow should be chained to the lecture-table, and enjoin his assistant to see that that was done without an hour's delay!
On one occasion, after working out part of a calculation on the long fixed blackboard on the wall behind the table, his chalk gave out, and he dropped his hand down to the long ledge which projected from the bottom of the board to find another piece. None was just there; and he had to walk a step or two to obtain one. So he enjoined McFarlane, his assistant, who was always in attendance, to have a sufficient number of pieces on the ledge in future, to enable him to find one handy wherever he might need it. McFarlane forgot the injunction, or could not obtain more chalk at the time, and the same thing happened next day. So the command was issued, "McFarlane, I told you to get plenty of chalk, and you haven't done it. Now have a hundred pieces of chalk on this ledge to-morrow; remember, a hundred pieces; I will count them!" McFarlane, afraid to be caught napping again, sent that afternoon for several boxes of chalk, and carefully laid the new shining white sticks on the shelf, all neatly parallel at an angle to the edge. The shelf was about sixteen feet long, so that there was one piece of chalk for every two inches, and the effect was very fine. The class next morning was delighted, and very appreciative of McFarlane's diligence. Thomson came in, put up his eye-glass, looked at the display, smiled sweetly, and, turning to the applauding students, began his lecture.
From time to time there were special experiments, which excited the interest of the class to an extraordinary degree. One was the determination of the velocity of a bullet fired from a rifle into a Robins ballistic pendulum. The pendulum, consisting of a massive bob of lead attached to a rigid frame of iron bars turning about knife-edges, was set up behind the lecture-table, and the bullet was fired by Thomson from a Jacob rifle into the bob of the pendulum. The velocity was deduced from the deflection of the pendulum, its known moment of inertia about the line of the knife-edges, the distance of the line of fire from that line, and the mass of the bullet.