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the direction of the arrow M. The moment of this force has its minimum effect when acting at F in the direction FR'. At any other point, d or d', the force is as gd' or g'd', that is as the sine of Ad or Bd', the polar distance of the point.
When the particle which receives the impulse at A arrives at B, it tends to move in the direction BR" parallel to AR, and therefore tends to restore the axis, from its position ab, to its previous position AB.
Now if the particle when at F, tending to move in the direction FR' by the impulse received at A, could move to H through the semicircle FBH in a time infinitely short, it would be equivalent to two equal forces at F and H, at equal distances from the fulcrum C of the lever FH, acting in lines FR’, HR", parallel to each other, and FH would therefore remain at rest, that is, the sphere would not rotate about the axis AB in the direction of the arrow M. But the particle cannot be conveyed from F to H in a time infinitely short; therefore the particle, in moving through the whole arc AFB, tends to make the sphere rotate about the axis AB in the direction of the arrow M.
Again, the impulse applied at A, in the direction AR, causes B to move in the direction BQ; when the particle at B is carried by the rotation of the sphere to H, it tends to move in the direction HQ', and therefore tends to turn the sphere about the axis AB in the direction of the arrow M', the whole sphere will therefore revolve about the axis AB in the direction of the arrows MM'.
Now, if instead of a single impulse acting at A, there is a constant force acting in a fixed direction AR, the rotation of the sphere about AB will continue until the axis CL is brought to coincide with CH, the sphere will then revolve about one axis HF. The force at A acting in the direction AR, which produced its minimum effect on the line of particles AB, when the sphere revolved about the axis KL, will now produce its maximum effect upon the same line of particles when the sphere revolves about the axis HF, and the sphere will revolve with an accelerating velocity.
From this reasoning it seems to follow, that if any sphere ADBC, Fig. 6, revolving about an axis AB, with a velocity however great, is acted on by a constant force, however small, tending to make it revolve about the axis CD, the sphere will revolve about a third axis perpendicular to both AB and CD, until AB is brought to coincide with CD.
If the force at A (Fig. 4), instead of acting in a fixed direction in space AR, acts always in a line at right angles to the plane of the lamina AFBH, the sphere will continue revolving about the axis AB in the direction of the arrows MM'. The oscillatory motion of the axis AB, compounded with this rotation, will cause the pole to describe an undulating line about the circumference of a small circle, whose diameter will depend on the intensity of the disturbing force applied at A.
II. APPLICATION OF THE PRINCIPLE TO PLANETARY MOTIONS.
Let S (Fig. 7) be the sun, and AMBM' the earth. Let the whole sphere AMBM' be divided into laminæ AB, &c., whose planes are parallel to each other, and at right angles to SC. If the whole of the matter of each lamina be collected into its centre of gravity, the equilibrium of the system will remain unchanged; the sphere will thus become a line of material particles M'M, whose centre is C.
Now, because the particle at M' is nearer to S. it will be more strongly attracted than an equal particle at M. The same may be said of every pair of particles equidistant from C, hence the centre of gravity G, of the whole line M'M, will always be nearer than its centre C to the sun S.
The point C is the centre of gravity of the line M'M when the attraction of the sun is disregarded; but G is the centre of gravity of the same line M'M when the attraction of the sun is considered.
Let the extremity M of the line M'M receive an impulse in the direction MN, at right angles to SM. In obedience to this impulse M will begin to move about the centre of gravity G. Let the point M move through an indefinitely small arc Mm. Then Gm is not greater than GM. But SG, Gm, are together greater than Sm; therefore SM is greater than Sm; therefore the particle at M has approached S in moving from M to m.
But if the line MM move in an orbit about the centre S, the centrifugal force generated by the orbitual motion acting on the particle at M, will tend to make it move in the arc MN, and therefore resist the approach of the particle to S, and therefore resist the motion of the line MM about the centre of gravity G. The same may be said of every particle between G and M; but the centre of gravity C of the whole line MM always lies between G and M; therefore the orbitual motion of the line MM about the centre S, resists the rotatory motion of the same line MM about the centre of gravity G. But the line MRM is made up of the centres of gravity of the laminæ which compose the whole sphere; hence the rotation of the sphere about the centre of gravity G is resisted by the orbitual motion of the sphere about the centre S.
If it be objected that M'M will not move about the centre G, but about the centre C, then let the circle GKL (Fig. 8), be the locus of the point G; and because CM is equal to Cm, therefore SM is equal to SC, Cm. But SC, Cm are greater than Sm: the rest of the argument follows as before.
If a sphere (Fig. 9), having no independent axial rotation, moves in an orbit about a centre of attraction S, having the same side CBD always turned towards S, it is obvious that it turns once round an axis CD in going once round its orbit, and that CD is perpendicular to the plane of its orbit. But if the sphere also rotates on an axis AB inclined to the plane of the orbit, whilst any force, however small, tends to make it rotate about the axis CD, then AB will gradually approximate to CD, and finally coincide with it.
III. PROGRESSIVE CHANGE IN THE FORM OF THE EARTH.
If a spheroid of equilibrium, NQSE (Fig. 10), contract uniformly in lines perpendicular to its surface, a new spheroid, N'Q'S'E', is produced of a greater degree of excentricity. For if CQ be the semi-transverse, and CN the semi-conjugate axis of the elliptical section NQSE of the spheroid, and if from these there be taken the equals NN' and QQ', the remainder CQ' has to the remainder CN' a greater ratio than CQ to CN, therefore the excentricity of the spheroid is increased.
But if the spheroid, rotating on an axis, contract in size, its velocity of rotation will be increased: this increase of velocity would tend to increase its excentricity. Now, it is barely possible that the increased excentricity due to the contraction, should be precisely the same as the increase of excentricity due to the increased speed of rotation; in that case the spheroid would still be one of equilibrium, notwithstanding the alteration of its form.
But if the increase of excentricity arising from contraction, is greater than that arising from the increased speed, the effect will be an accumulation of matter in the equatorial region, an increase of pressure on the internal mass, and a tendency to subsidence into the northern and southern hemispheres. A change of form is then necessary to restore equilibrium. This may not take place uniformly per gradum ; for if there be a resistance from a
rigid external crust, the force must accumulate until it exceeds the resistance, and thus frequent adjustments, per saltem, may ensue. It is probable, therefore, that the earth's form is undergoing a slow progressive change ; and if so, it must be taken into account by those who venture to speculate on the causes of earthquakes, volcanic eruptions, and the upheaval of mountains.
IV. DESCRIPTION OF THE GYROSCOPE. 1. When the disc is in rapid motion it resists any force which tends to alter its plane of rotation, because the point at which the force is applied is conveyed to the point diametrically opposite, before it has time to move in any perceptible degree in the direction of the impressed force.
2. When a force is applied to turn the disc about a vertical axis, whilst it is turning on a horizontal axis, the plane of rotation is changed. Assume a point in the upper edge of the disc. It tends to move in the direction of a tangent, and when the disc is moved about a vertical axis, the point assumed still tends, by its inertia, to continue its motion in the same direction as before, that is, now obliquely to the disc, and therefore tending to alter its plane of rotation.
3. When a weight is suspended to the extremity of the horizontal axis on which the disc turns, the plane of the disc is not sensibly drawn out of the perpendicular, but a motion is produced about the vertical axis. Assume a point in the upper edge of the disc. Place a thread on it by means of a piece of sealing wax, leaving the two ends free from the point of attachment. Pull one thread in the direction of the tangent, thus producing a rotation of the disc: pull the other at right angles to the plane of the disc, thus producing the same effect as the weight. This force cannot perceptibly alter the plane of rotation,