(Vol. iv., pp. 129. 177. 235. 277.)
Since my last communication on this subject (Vol. iv., p. 235.) I have
been engaged in examining the theory, and the experiments connected with
it, somewhat more closely; and, in the meanwhile, I abstain from
replying to the last observations of A. E. B. (Vol. iv., p. 277.)
A. E. B. says it was "uncourteous" in me to call the theory which he put
forward _his_ theory. I beg pardon for the offence. I intended by the
expression merely to indicate the particular theory which he advocated.
I believe its author is M. Chesles. The theory in question is:
"That the variation of the pendulum's plane is due to the excess
of velocity with which one extremity of the line of oscillation
may be affected more than the other."
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.