Note relating to M. Foucault's new mechanical proof of the Rotation of the Earth
The following papers were read: - 1. "Note relating to M. Foucault's new mechanical proof of the Rotation of the Earth." By C.Wheatstone, Esq., Corresponding member of the Acadamies of science of Paris, Berlin, Brussels, Turin, Rome, Dublin, &c. Reveived May 15, 1851.
The experiment which led M. Foucault to his ingenious and interesting researches relating to the motion of the earth, is stated by him thus:- "Having fixed on the arbor of a lathe and in the direction of the axis, a round and flexible steel rod, it was put in vibration by deflecting it from its position of equilibrium and leaving it to itself. A plane of oscillation is thus determined, which, from the persistence of the visual impressions, is clearly delineated in space; now it was remarked that, on turning by the hand the arbor which serves as a support to this vibrating rod, the plane of oscillation is not carried with it."
This persistence of the plane of oscillation of a vibrating rod, notwithstanding the rotation of the point to which its end is fixed does not appear to have hitherto been made the subject of philosophical observation. Ordinary notions even seem to have been opposed to this now recognized fact. Chladni in his treatise on Acoustics, in the chapter, "On the co-existence of vibrations with other kinds of motion." states as follows:-
"Vibratory motion may co-exist with all other kinds of motions in an infinity of different manners, as has been demonstrated by Dan. Bernouilli, and L. Euler in vols. xv and xix, of the Nor. Comment. Acad. Petrop., and confirmed by experiment. These co-existences of different motions occur in all sonorous bodies without exception; we may, for example, produce the sound of a string stretched on a board, or that of a plate, a tuning fork, a bell, &c; and while the vibrations still last, impress on this sonorous body a motion of rotation round its axis, and at the same time a progressive motion: thus all these motion may be performed in the same time, without one being hindered by the other; but the absolute motion of each point will be very complicated."
Now this is true only when the vibrating body is constrained to vibrate in one direction. When the rod or string is equally flexible in every direction, the plane of vibration given to it from any original impulse is constantly maintained whatever may be the velocity of rotation communicated to its point of support, provided the axis of vibration remains in the same position, or move only parallel to itself.
This observed independence of the plane of oscillation on the point of attachment led M. Foucault to assume, that were a flexible pendulum suspended from a fixed point in the prolongation of the axis of the Earth, that is above the plane of oscillation maintaining an invariable position in space would appear to a spectator on the earth's surface and moving with it to make an entire revolution in twentyfour hours, but in the opposite direction to that of the rotation of the Earth.
What takes place at other points of the earth's surface is more difficult to determine; but M. Foucault, from mechanical and geometrical considerations, was led to the conclusion that the angular displacement of the plane of oscillation is equal but opposite to the angular momentum of the earth multiplied by the sine of the latitude. According to the theory of rotation, first established by Frisi and more fully developed by Euler and Poinsot, the velocity of rotation of the earth may be considered as the resultant of two angular velocities, one round the vertical of the point where the observer is placed, and the other round the meridian or horizontal line lying N. and S. The component of the angular velocity estimated round the vertical axis is n sin gamma, and the plane of oscillation not participating in this motion remains at rest with respect to it, and therefore appears to an observer moving with the point, to rotate with the same velocity in the contrary direction.
The experiment made by M. Foucault is said, both in the direction and magnitude of the motion of the plane of oscillation of the pendulum, fully to conform the indications of the theory. The difficulty, however, of the mathematical investigation of the subject, and the delicacy of the experiment, liable as it is to so many extraneous causes of error, have induced many persons to doubt either the reality of the phenomenon or the satisfactoriness of the explanation. Another experimental proof, therefore, not depending on the rotation of the earth, that the plane of oscillation of a vibrating line remains at rest with relation to the vertical component of the real axis of rotation, may not be unacceptable. With this in view I have devised the apparatus I am about to describe.
A semicircular arch from one one to two feet radius is fixed vertically on a horizontal wheel, and may thus be moved with any degree of rapidity from any one azimuth to another. A rider slides along the inner edge of the arch, which is graduated, and may be fixed at any degree marked thereon. A spiral spring wire, by means of which a slow vibration is obtained with a comparatively short length, is attached at the lower end to a pin fixed in the axis of the semicircle, so that the point of attachment may be the axis of rotation, and at the upper end it is fixed to a similar pin in a parallel position fixed to the rider. The vertical semicircle is not placed in a diameter of the horizontal wheel, but parallel to it, at such distance as not to intercept, from the eye of the observer, the vertical plane passing through the diameter, and in which plane the wire in all its positions remains.
When the upper end of the wire is placed at 90º, that is when it coincides with the axis of rotation, if the wire be caused to vibrate in any given plane, say from N. to S., it will continue to do so whatever rotation may be communicated to the wheel; so that with respect to the moving wheel, or the axis of the wire, the plane of vibration will move with the same velocity and in the opposite direction. When the rider is fixed at 30º, and the wire makes therefore an angle of 60º with the axis of rotation so as to describe in its motion the surface of a cone having this inclination to the vertical, it will be observed that the plane of the vibration makes one complete rotation during two rotations of the wheel; this is best observed by fixing the eye so that its axis shall coincide with a line in the same vertical plane with the wire, while walking round with the wheel during its rotation. When the rider is fixed at 19½º, the plane of vibrations makes one rotation during three rotations of the wheel; when fixed at 14½º, it makes one rotation during four of the wheel, &c.; and when it is fixed at 0º, the wire lying horizontally, no rotation of the plane of vibration occurs. It is needless to observe that the sines of 90º, 30º, 19½º, 14½º, 0º, correspond to the numbers 1, 1/2, 1/3, 1/4, 0, the reciprocals of the numbers expressing the respective times of rotation. 
It is not necessary that the wire should have one of its ends fixed in the axis of rotation: if it be parallel to a wire so fixed, the rotation of the plane of vibration will be exacly similar; in such a case the wire or axis of vibration will describe the surface of two cones having their common apex in the axis of rotation.
The axis of a flexible pendulum can only assume a position vertical to the point of the earth's surface over which it is placed. Were it possible to maintain the vibration of a stretched wire occasioned by an original impulse, for a sufficient length of time, the apparent rotation of its plane of vibration would vary with the inclinations of the wire to the axis of the earth: placed in this axis, it would make a rotation in 24 hours, it would become progressively slower according to the law above given, as it approaches the plane of the equator, and when anywhere in this plane the vibrations would always be performed in the same direction.
- When the dimensions of the apparatus are as above given, I find that hardened brass wire (no 26), coiled so as to form a helix of one quarter of an inch in diameter, shows the effect well. The thickest spiral wire employed in the manufacture of artificial flowers, which can be procured of any wire-drawer, will also answer the purpose.
The best way of setting the wire in vibration is to press the finger upon it in the middle, so as to deflect it in the plane in which the vibrations are required to continue, and then suddenly to withdraw the finger in the direction of the vibrations. The deflection must not be too great, or the elasticity of the wire will be injured.
This work published before January 1, 1923 is in the public domain worldwide because the author died at least 100 years ago.