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APPENDIX.


I.

FOUCAULT'S PENDULUM EXPERIMENT.


In the year 1851, a method of rendering certain effects of the earth's rotation visible to the eye was made known by M. Foucault, who had been led to discover it by considering how the rotation of the earth ought to affect the apparent motion of a pendulum vibrating freely at the earth's surface.

If a heavy body, as for example, a sphere of metal be suspended by a string from a point A, Figure 66, vertically above N, the North Pole of the earth, and allowed to hang freely, the motion of the earth about its axis ANS will twist the string, and so cause the sphere to rotate about its vertical diameter. If, now, the sphere be drawn aside to a point B and allowed to drop gently, it will begin to vibrate in the plane NAB, and as the rotation communicated to the sphere does not tend to withdraw it from that plane, it will continue constantly to move in it. A spectator near N, partaking of the earth's motion, changes his position with reference to this fixed plane: but being unconscious that he is moving himself, he attributes to the fixed plane a motion exactly similar to his own, but in the opposite direction. To him it will therefore appear to revolve from east to west about the line NA, making a complete revolution in the course of a day. At the South Pole a similar appearance would be observed.

Popular astronomy airy(1881) page 278 figure 66.png

Fig. 66.

At places situated elsewhere on the earth's surface, it is less easy to anticipate the result; but some idea of the effect produced on the plane of vibration may perhaps be conveyed by the following explanation.

It has been remarked above, (page 108,) that a single force, acting in a given direction, may be resolved into two forces acting in given directions: and that these two forces acting together may be regarded as producing the same effect as the single force acting alone. In like manner a single motion of rotation about a given axis may be resolved into two motions of rotation about two given axes: and if these two motions take place simultaneously, they may be regarded as together producing the same effect as the single motion. Thus, if a body (which for the sake of simplicity we may suppose to be

Popular astronomy airy(1881) page 279 figure 67.png

Fig. 67.

spherical) be made to rotate about the line OA, Figure 67, any point in it will describe a circle in a plane perpendicular to OA, with its centre on that line. Suppose that in a given time the point P is thus brought from one position P to another R; then it is possible to produce the same change in position by giving the body two successive rotations about two lines, OB, OC. For by virtue of a rotation about the line OB, P may be made to describe the arc PQ of a circle having its centre on the line OB, and thus be brought to the position Q. Again, by a rotation about OC, Q may be made to describe the arc QR of a circle, having its centre on the line OC, and be brought to the position R. Thus, by two properly chosen steps, P to Q and Q to R, P is brought to R; and by giving the rotations about OB, OC, proper degrees of rapidity, each of the two steps may be taken in the same time as was the single original step, P to R. Suppose, now, that the two rotations, which we have hitherto supposed to be made one after the other, are made simultaneously: then the steps will, as it were, be taken together and in the same time as the single step; and P will, under the combined influence of the two rotations, be brought to the same position as it would be brought to under the influence of the single rotation. Thus, two rotations, about OB, OC, of proper degrees of rapidity, may be regarded as producing the same effect on every point of the body as does the single rotation about OA.

Now, if Q, Figure 66, be a place on the Equator, the earth has no motion of rotation about A'QE; and consequently the plane of vibration, having itself no motion of rotation, will appear to preserve a fixed position relatively to an observer at Q. Again, let NOS, Figure 68, represent the earth's axis; let O be the centre of the earth and P a place on its surface; and let OR be a line perpendicular to OP and in the plane NPS, or the plane of P's meridian. Then, instead of supposing the earth's motion to consist of a single rotation about NOS, we may regard it as arising from two properly chosen rotations about OP and OR; and we may consider separately what would be the apparent effects of these rotations on the pendulum: just as at page 111, the effects of the two components of a force are separately considered.

Popular astronomy airy(1881) page 281 figure 68.png

Fig. 68.

If, then, the earth rotated about OP alone, P would be one of its poles; and therefore, as has been already explained, to a spectator near P the plane of vibration would appear to revolve in the same time as the earth would revolve about OP alone, but in the opposite direction. Again, if the earth revolved about OR alone, P would be situated on the Equator; and a spectator would not observe any change in the plane of vibration. Thus at every place not situated on the Equator, the plane of vibration will appear to change its position; and such change will be entirely due to the component rotation about OP.

It is not difficult to submit these conclusions to an experimental test, and this has actually been done at many places. Care must be taken not to give the sphere any lateral motion at the moment when it is dropped ; otherwise, (referring to Figure 66,) it would at once be carried out of the plane BAN, and the previous reasoning would no longer hold. In practice, this object is attained by fastening the sphere to a wall or other fixed body by means of a second string, so as to keep it in such a position as AB ; and then setting the second string on fire at an instant when the sphere is observed to be conpletely at rest.


II.

THE GYROSCOPE.


In general, a body which is set rotating about a line will not continue to revolve about it; on the contrary, the axis of rotation will usually change its position relatively to the body. Whatever be the shape of the body, however, it is always possible to find a line fixed relatively to the body, such, that the body having once begun to rotate about it will continue to do so, provided no external force act: such a line is called a permanent axis of rotation. If the body be perfectly symmetrical about any line within it, that line is a permanent axis; thus the diameter of a sphere, the diagonal of a cube, and the axis of revolution of a spheroid, (page 60,) are permanent axes of the sphere, cube, and spheroid. If, further, when a body has once been made to rotate about a permanent axis, no force intervene to alter the original motion, the axis will always preserve the same direction in space, as well as the same position in the body, a circumstance alluded to in the lectures, (page 78.)

Suppose, then, a body to be set rotating about a permanent axis and to be mounted so that the force of gravity does not interfere with the rotation; suppose, also, that at the commencement of the motion the axis points to some star; then, if it be true that the star does not move, the axis will always point to it so long as the rotation lasts. If it be true that the apparent diurnal motions of the stars are due to an actual motion of the earth; then to an observer on the earth's surface, the axis will appear to move so as to follow the star. But if, on the other hand, the star move while the earth remains at rest, no such apparent change in the position of the axis will be observed. Thus we have the means of testing, by direct experiment, the truth of the conclusion arrived at in lecture II, (page 78,) that it is the earth which revolves and not the stars.

The following is the description of an instrument, contrived by M. Foucault, for the purpose of making such an experiment as we have just mentioned. DD', Figure 69, is a heavy metallic disc, mounted on an axis which passes through O, the centre of the disc, and is perpendicular to its two sides. The extremities of this axis terminate in pivots CC', which fit into holes made at opposite extremities of the diameter of a circular ring BCB'C', which is furnished with two knife edges (similar to those of a balance) at B and B', and so arranged that BB' is the diameter of the ring perpendicular to CC'. The knife edges rest in holes made at opposite extremities of the horizontal diameter of a vertical circle ABA'B', which suspended by a fine wire SA from the fixed point S. At A', the opposite extremity of the vertical diameter AA', is a pivot,

Popular astronomy airy(1881) page 284 figure 69.png
Fig. 69.

which rests in a small hole. All the pivots are carefully polished, so that friction may be avoided as much as possible; and the dimensions of the different parts of the instrument are so adjusted that is the common centre of the disc and the rings.

Now, as the disc is symmetrical about CC', that line is a permanent axis of rotation; and as O is the common centre of gravity of the different parts of the machine, and is supported by the string SA and the reaction at A', the force of gravity will not interfere with the rotation. Thus the instrument satisfies all the conditions necessary for making the experiment. It is also clear, that the axis CC' may be placed so as to point in any direction we please by moving first the ring ABA'B', and then the ring BCB'C' into proper positions.

To make the experiment, BCB'C' is removed from its supports, and a rapid motion of rotation is impressed on the disc. The ring BCB'C' is then restored to its place. It is found, that if at any instant CC' points to a fixed star, it continues to do so while the disc rotates, and thus appears to an observer to change its position relative to the surface of the earth; unless, indeed, the star be the pole star, in which case the observer will not notice any apparent change in its direction.

This instrument is called the "Gyroscope."


III.

THE DENSITY OF THE EARTH


It is stated in the lectures, (page 222,) that if the law of universal gravitation be true, it is found (by a difficult mathematical investigation) that the attraction of the whole earth, considered as a sphere, on a body at its surface is the same as if the whole matter of the earth were collected at its centre. It is also found that the attraction of the earth on a body within its surface is the same as if the spherical shell situated between the body and the earth's surface were removed; or is the same as if all the matter situated nearer to the earth's centre than the body were collected at the centre, and all the matter situated at a greater distance were removed.

If the earth were of uniform density throughout, it would follow from these propositions that the force of gravity at the bottom of a mine would be less than the force at the top. To show this, suppose that the mine reached half-way to the centre of the earth. Then (since the volumes of spheres vary as the cubes of their diameters) the quantity of matter nearer to the earth's centre than the bottom of the mine would be only one-eighth of the whole quantity of matter in the earth. But the attraction of a quantity of matter at the earth's centre would be more powerful on a body at the bottom of a mine than on one at the top, in the inverse ratio of the squares of the distances of the bodies from the earth's centre: that is in the present case in the ratio of four to one. Hence the attraction on a body at the bottom of the mine would be, on the whole, less than the attraction on a body on the top in the. ratio of one to two.

If, however, the earth be not of uniform density, but its density increase towards the centre, then though the attracting mass which acts on a body at the bottom of a mine be smaller, yet the diminution in the force of gravity so occasioned may be more than compensated by the comparative nearness of the attracted body to the denser parts of the earth. From the two laws of the attraction of spheres, which have been stated above, it is possible to calculate the ratio which the force of gravity at the bottom of the mine would bear to that at the top, on any supposition we choose to make as to the ratio which subsists between the mean density of the earth and the density of the surface; so that if we know one of these ratios we can immediately infer the other. Now, pendulum observations afford us the means of determining the force of gravity at any place, (page 248,) and therefore, if the times of vibration of a pendulum at the top and bottom of a mine be found, the ratio of the force of gravity at the top to that at the bottom may be calculated, and thence the ratio of the mean density of the earth to that of its surface.

This mode of determining the mean density was put in practice by the Astronomer Royal, at the Harton Coal Pit, near South Shields, in the year 1854. The mean density deduced from his observations is 6·565: a value considerably exceeding that found from the Schehallien and Cavendish experiments.

THE END.