ART. XXII.-Foucault's Pendulum Experiment. THE peculiar interest of Foucault's experiment with the Pendulum, by which the rotation of the earth is proved even within the walls of an ordinary lecture room, has obtained for it very general attention, and it has already been repeated in numerous places in this and other countries. The occurrence from which M. Foucault was led to his discovery is thus related by him.* "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 around with the hand, the arbor which formed the support of this vibrating rod, the plane of oscillation was not carried with it, but always retained the same direction in space." From this came the conclusion that a pendulum set in motion will continue in the same plane of vibration, however the point of suspension be rotated-a fact easily proved by a simple trial with a weight at the end of a cord. The rotation of the point of suspension may make the pendulum revolve on its axis; but the plane of vibration will remain the same. The reason for this is obvious: the swinging pendulum, when about to return (after an outward swing) from its point of rest, is made to move from that point by gravity alone, and can therefore fall but in one direction; and the momentum acquired by falling carries it beyond this centre in the same direction to the point of rest on the other side; here again it is in a like condition, and must return under the force of gravity in one and the same line, gravity acting in the same direction whether the point of suspension be rotated or not. Thus the plane of vibration is fixed from the very nature of the forces at work. It is evident, therefore, that if a pendulum were swinging at the pole of the earth, the plane of vibration, as it would not change with the revolution of the earth, should mark this revolution by seeming to revolve in the contrary direction, and in 24 hours it would make apparently the whole circuit of 360 degrees. But at the equator, the plane of vibration is carried forward by the revolution of the earth, and so undergoes no change with reference to the meridians. Between the equator and poles, the time required for the pendulum to make 360° varies according to the latitude, being greater the farther from the pole. For these intermediate latitudes the problem is beautifully solved on a globe. Having a globe for the trial, select, for instance, latitude 30°, and apply the principle that the plane of vibration is constant * Bibl. Universelle, March, 1851. notwithstanding the revolution. Suppose the plane of vibration to be the meridian of Greenwich: draw a series of lines across the parallel of 30°, parallel to this meridian; each of these lines will show precisely the position of the pendulum, in the revolution, at the point where the line intersects the parallel of latitude. If, for example, we draw one of these parallel lines across the intersection of 30° with the first meridian circle east of Greenwich; another across the intersection of the parallel of 30° with the second meridian; another across the third, and so on ;-each will mark the position of the plane of vibration after a revolution of as many degrees as are included between the meridians. Continuing this on, the parallel lines become more and more easterly, and finally they are exactly east and west in longitude 180°; or in other words, the plane of vibration, which is still parallel to the Greenwich meridian, is actually at right angles to the meridian of 180°; this plane has therefore changed apparently 90°, while the earth was revolving 180° or making half its circuit. And if this be continued, it will be found that the parallel lines will make the whole circuit of 360° in twice the earth's revolution, or what is equivalent, in 48 hours. These parallel lines drawn on the globe, it is to be observed, should be parts of great circles, and are strictly parallel to one another only at the intersections with the parallel of latitude, like the meridians where they cross the equator; their tangents at these intersections will be actually parallel throughout, like the tangents to the meridians at the equator, and these tangents represent the true angle of the plane of vibration. Again take latitude 70°, and in the same manner draw lines parallel to the meridian of Greenwich, going eastward. These parallel lines will rapidly diverge from the meridian, the divergence being nearly 9 degrees for every 10 degrees of longitude; in lon. 95° 46' E. the intersecting line will point east and west, having made 90° in its apparent revolution. In 191° 33′ it will have made 180°, and in 383° 6', (or 23° 6' east of the meridian of Greenwich,) it will have completed the whole 360°; so that in latitude 70° the time of the apparent revolution of the plane of vibration, is a little over 254 hours. The globe thus exhibits to the eye the actual uniform position of the plane of vibration, and at the same time its apparent revolution. By marking these parallel lines at one end with an arrow-head, and also describing an arc with the point of intersection as a centre, the increasing divergence from the meridians, and the apparent rotation by east, south, and west to north, is well shown. If now tangents be supposed to be drawn to each meridian circle, at its intersection with any parallel of latitude, say that of 70°, these tangents will intersect at point in the axis of the earth extended; and the angle included between any two such SECOND SERIES, Vol. XII, No. 35.-Sept., 1851. 26 tangents is the angle which the corresponding meridians make with one another in the given latitude. The plane of vibration, since it continues parallel to itself, will therefore change with reference to the meridians, just the amount of the angle included between these tangents. This is readily seen on the globe, marked off as above explained, in which the Greenwich meridian is the starting point. The angle which one of the parallel lines drawn on the globe (or its tangent) makes with the meridian it intersects is (from the nature of parallel lines) equal to the angle between the tangent to this meridian and that of Greenwich. The sum therefore of the angles between all the tangents to the meridians, will be the amount of apparent variation in the plane of vibration for 24 hours. These tangents-which are strictly the cotangents of latitude-form together the surface of a cone whose base is the parallel of latitude, and whose angle of surface at summit is equal to the revolution of the pendulum in 24 hours. If this cone be supposed to be cut open and laid out on a flat plane, it will form a sector of a circle, whose angle at the centre equals the angle around the apex of the cone. The radius of this sector (or of its circle) equals the cotangent of latitude, (cot L); and the circumference of the sector is actually a parallel of latitude-a parallel of latitude having been the base of the supposed cone. Now as the number of degrees in any arc varies with the length of the arc arc as related to the radius of the circle, or as ; and since the arc R here is a parallel of latitude and these parallels vary with the cosine of latitude (cos L), and since also the radius (R) equals cot. L -it follows that the number of degrees in the arc (or the number expressing the apparent motion of the plane of vibration for 24 cos L hours,) will vary as of latitude. cot L' an expression equivalent to the sine Other demonstrations arrive at the same result, but none is more simple or more conclusive. The theory thus involves no necessary consideration of the forces engaged, by which many explanations of the experiment are encumbered, but is simply a geometrical problem, based on the position of the meridians of a revolving sphere, and the fact that the pendulum moves in a fixed direction, parallel to the meridian in which it is started. The idea involved in Foucault's experiment "seems to have occurred long ago, and is mentioned in a paper in the Phil. Trans. 1742, No. 468, by the Marquis de Poli, in the course of some observations on the pendulum of a different kind. He remarks, 'I then considered (adopting the hypothesis of the earth's motion,) that in one oscillation of the pendulum there would not be de scribed from its centre perfectly one and the same arc in the same plane'; but he does not pursue the subject as being foreign to his immediate object."* It appears also, (Comptes Rendus, 1851, No. 6,) that in 1837, Poisson had hinted at such an effect, but supposed it of insensible amount. Different modes of varying the experiment have been proposed. The following is suggested by Poinsot. If two balls are suspended in contact, with a spring between them, and then by a sudden action of the spring are thrown apart and thus held by some attached contrivance, they will retain, after the change of position, their original rate of velocity, or that of the earth where they originally were, and consequently will commence to revolve. It would seem that if the cause mentioned be the source of motion they should not revolve if thrown apart east and west, or at right angles to a meridian. But if viewed in a different light it appears to be a fact that they would revolve in whatever direction separated. Suppose the halves of a sphere to be thus thrown apart and so retained; they have the momentum of the earth as before, but no relative motion, and consequently they are in the condition. of the opposite points of rest of a vibrating pendulum, and will have an apparent revolution around the centre between them. M. Baudrimont has observed that if torsion could be wholly avoided, a ball suspended or a horizontal bar would exhibit the rotation like a vibrating pendulum. But in these projects there is a fatal difficulty besides that from torsion. Such bodies, at the time when suspended, have the actual motion of objects at the place where they are, and will retain it; and consequently under no circumstances can they give results like those of the pendulum, or show any corresponding effect whatever. In performing the pendulum experiment, the ball of the pendulum should be an accurately turned sphere or cylinder, having a place of suspension above and an elongated point below, well centered. It is best suspended by means of wire. The simplest mode to fasten it is to solder the wire into a hole drilled of the same size in a larger piece of metal, and secure this last in the ceiling or other support above. The swinging of the pendulum, by flexing the wire at the junction above, will after a while break it off. But if of steel instead of iron, the flexion will be distributed along a portion of its length, and the wire will last longer without breaking. To mark the motion, a circle (three feet or more in diameter) divided into degrees, is placed below the pendulum; the apparent motion of the plane of vibration, is observed (in north latitude) to be uniformly from left to right, or with the hands of a watch. For starting the pendulum when there is no better contrivance, it may be held, previous to letting it swing, * Phil. Mag. [4] i. 561. by a cord passed once around it, the hand resting at the same time on some firm object; when all oscillation has ceased, it may be let off by dropping the cord. In going the circuit, the ball of the pendulum has always the same side towards the north, and consequently it rotates on its axis with the earth. However careful the trial, the motion of the pendulum soon appears to be somewhat elliptical. This has been attributed solely to accidental causes, and it is true that the slightest error of direction in letting off the pendulum will necessarily produce it. But on a following page, Rev. C. S. Lyman shows that a degree of ellipticity is a necessary result of the earth's rotation. J. D. D. ART. XXIII-Note on Heteronomic Isomorphism; by 3 IN Volume Nine of this Journal* I published views on certain isomorphous groups among minerals, tracing this quality to a relation in atomic volume, a principle already admitted, but showing that this relation is most correctly exhibited when the aggregate atomic volume is divided by the number of atoms (or molecules) of the elements present. Thus for Fe 0, I would divide the atomic volume, obtained in the usual way, by 2, which reduces the compound to the condition of a unit, as if consisting, as in effect it does, of FeO. Again, for Fe2 O3, I divided by 5, as the compound consists essentially of Fe3 O, the sum of the fractions making a unit. This method carried out with the feldspars, exhibited a relation between them of actual (or approximate) equality in atomic volume, and also a connection between the system of crystallization of these species and the atomic volume, the monometric (as Leucite) having the highest number, the monoclinic the next highest, and the triclinic a lower number. 3 Farther study is required before all the difficulties connected with this subject, arising from isomerism and differences in the elements, are mastered. But in the comparison of compounds consisting of like elements though in different proportions, the method appears to be satisfactory, and affords conclusions of great simplicity. These views are well elucidated by many groups of silicates, and they give increased interest to the recent results of Rammelsberg with the tourmalines. This distinguished chemist, after numerous analyses, makes out five chemical groups in * P. 220, 1850. |