I ventured to pronounce this to be untenable, and begged A. E. B. to "reduce it to paper." Upon this he remarked:
"H. C. K. is surely not so unphilosophical as to imagine that a theory, to be true, must be palpable to the senses. If the element of increase exist at all, however imperceptible in a single oscillation, repetition of effect must eventually make it observable. But I shall even gratify H. C. K., and inform him, that the difference in linear circumference between two such parallels in the latitude of London, would be about 50 feet; so that the northern end of a 10 feet rod, placed horizontally in the meridian, would travel less by that number of feet in twenty-four hours, than the southern end. This, so far from being inadequate, is greatly in excess of the alleged apparent motion in the place of the pendulum's vibration."
I think, if A. E. B. will reconsider this opinion, he will find that, so far from being "greatly in excess," it is inadequate to account for the amount of apparent motion of the plane of the pendulum. For the onward motion of the plane of a 2 sec. pendulum, describing a circle of 10 feet diameter in twenty-four hours, amounts to ·0087 inch at each beat; 50 feet will be the difference in the distance the two extremities of the arc of vibration will travel in twenty-four hours; that is, ·0138 inch in 2 seconds of time: but this is for a difference of 10 feet; therefore, for 5 feet, the distance from the centre, it is ·0069 inch; whereas the arc described is ·0087 inch, which is absurd.
However, there is another equally fatal objection to this theory, founded on experiment; to make which objection good, I will not merely adduce the result of my own, but that of certain experiments carried out at Paris, which place the matter beyond a doubt. In the Pantheon, at Paris, there is a pendulum of the length of 230 feet, by means of which experiments can be made under the most favourable conditions possible as regards suspension, exclusion of currents of air, &c. &c. While witnessing the trials that were being made, a relation of mine requested that the pendulum might be set to oscillate east and west; and the result was, that the arc described after an interval of ten minutes, was the same as that described when the pendulum was oscillating north and south.
To return to the original theory. I stated formerly that I had no faith in the experiments which had been published. I now repeat that I believe all the experiments that have been made, with the view of showing the rotation of the earth, and the independence of the pendulum of that rotation, are inconclusive; and for the following reason, the impossibility of obtaining perfect suspension. Even in a still atmosphere, and with a pendulum formed of the rigid rod and a "bob," the axis of both of which shall be precisely in a line with the point of suspension; yet, until suspension can be effected on a mathematical point, and all torsion and local attraction got rid of, the pendulum will not continue to swing in the same plane for many consecutive beats; because the slightest disturbance will cause the "bob" to describe an ellipse; and, by a well-known law, the major axis of that ellipse will go on advancing in the direction of the revolution. This advance is by regular intervals; and my belief, founded on my own experiments, is, that the astonished spectators at the Polytechnic Institution, while intently watching, as they believed, the rotation of the earth made visible, were watching merely a weight suspended by a cord, which, disturbed from the plane in which it was set to oscillate, was describing a series of ellipses on the table, very pretty to look at, but having no more to do with the rotation of the earth than the benches on which they were sitting.
At the same time, however, that I assert the inefficacy of any experiments with the pendulum as tending to show the earth's rotation, I admit that, provided a pendulum could be made to preserve its plane of oscillation for twenty-four hours, it would oscillate independently of the rotation of the earth, and actually describe a circle round a fixed table in that interval. The mathematical proof of this proposition is of a most abstruse nature; so much so, indeed, that it is understood to have been relinquished by one of our ablest mathematicians. But that it is likely to be true, and one not difficult to comprehend, I think I can show to A. E. B.'s satisfaction in a few lines.
If a pendulum be placed at one of the poles of the earth, it is obvious, that while it swings in one plane, the revolution of the earth beneath it will cause it to appear to describe a complete circle in twenty-four hours. This position is simple enough, but it is true also in any latitude, excepting near the equator. For there is no doubt, that, as gravity acts on the pendulum, only in the line which joins the point of suspension and the centre of the earth (thereby merely drawing the "bobs" towards that line) it can have no effect on the plane of oscillation; for the line of gravitation remains unchanged with respect to the pendulum, during a whole revolution of the earth on its axis. Take a map of a hemisphere, and on any parallel, say 60° of latitude, draw three pendulums, extended as in motion, with their centres of gravity directed toward the earth's centre, one on each extremity of the parallel of latitude, and one midway between the two; extend the "bobs" of the first two north and south, and those of the middle one east and west. Number them 1, 2, and 3, from the westward. It will then be observed that the plane of oscillation of the three pendulums, thus placed, is one and the same—that of the plane of the paper; and moreover, that the lower "bob," which is south at No. 1., is west at No. 2., and north at No. 3. By this it will be evident, that the revolution of the pendulum will be through the whole circle, or 360° in twenty-four hours, at all points of the earth's surface, excepting near the equator; the line joining the "bobs" remaining in a parallel plane.
I say, excepting near the equator; for it will be seen on looking closely at the above illustration (which would be better on a globe) that the three pendulums are not strictly in the same, or even a parallel plane; inasmuch as the plane of oscillation must pass through the point of suspension, and the centre of the earth. But still the pendulum has a tendency to remain in a parallel plane, as nearly as the figure of the earth will allow,—the chord of the arc of oscillation remaining in a plane parallel to itself. It will be seen that, as we approach the equator, the plane of oscillation is forced from its parallelism more and more, until, on the equator, it has no tendency to return, as all planes are there the same with reference to the centre of the earth.
I may add that there is a variation of the above theory, which has found many advocates, viz. that the pendulum will make the complete revolution in a period varying from twenty-four hours at the poles, to infinity at the equator; varying, that is, as the sine of the latitude. This seems, à priori, not so likely as the former, while it equally wants mathematical proof.
H. C. K.