motion of a

required. But by § 79 this may be effected by rotation about General a certain axis perpendicular to the plane S, unless the motion rigid body. required belongs to the exceptional case of pure translation. Hence [this case excepted] the body may be brought from the first position to the second by translation through a determinate distance perpendicular to a given plane, and rotation through a determinate angle about a determinate axis perpendicular to that plane. This is precisely the motion of a screw in its nut.

103. In the excepted case the whole motion consists of two translations, which can of course be compounded into a single one; and thus, in this case, there is no rotation at all, or every plane of it fulfils the specified condition for S of § 102.

Rotation.

104. Returning to the motion of a rigid body with one point Precessional fixed, let us consider the case in which the guiding cones, § 99, are both circular. The motion in this case may be called Precessional Rotation.

The plane through the instantaneous axis and the axis of the fixed cone passes through the axis of the rolling cone. This plane turns round the axis of the fixed cone with an angular velocity (see § 105 below), which must clearly bear a constant ratio to the angular velocity w of the rigid body about its instantaneous axis.

105. The motion of the plane containing these axes is called the precession in any such case. What we have denoted by is the angular velocity of the precession, or, as it is sometimes called, the rate of precession.

The angular motions w, are to one another inversely as the distances of a point in the axis of the rolling cone from the instantaneous axis and from the axis of the fixed cone.

For, let OA be the axis of the fixed cone, OB that of the rolling cone, and OI the instantaneous axis. From any point P in OB draw PN perpendicular to OI, and PQ perpendicular to OA. Then we perceive that P moves always in the circle whose centre is Q, radius PQ, and plane perpendicular to OA. Hence

A

Q

I

N

B

Rotation.

Precessional the actual velocity of the point P is NQP. But, by the principles explained above, § 99, the velocity of P is the same as that of a point moving in a circle whose centre is N, plane perpendicular to ON, and radius NP, which, as this radius revolves with angular velocity w, is wNP. Hence

I. Convex cone rolling

on convex.

cone.

Q.QP=w. NP, or w: :: QP: NP.

Let a be the semivertical angle of the fixed, ẞ of the rolling, Each of these may be supposed for simplicity to be acute, and their sum or difference less than a right anglethough, of course, the formulæ so obtained are (like all trigonometrical results) applicable to every possible case. We have the following three cases :

II. Convex cone rolling inside con

cave.

III.Concave cone rolling outside convex,

Cases of precessional rotation.

106. If, as illustrated by the first of these diagrams, the case is one of a convex cone rolling on a convex cone, the precessional motion, viewed on a hemispherical surface having A for its pole and O for its centre, is in a similar direction to

cessional

that of the angular rotation about the instantaneous axis. Cases of preThis we shall call positive precessional rotation. It is the case rotation. of a common spinning-top (peery), spinning on a very fine point which remains at rest in a hollow or hole bored by itself; not sleeping upright, nor nodding, but sweeping its axis round in a circular cone whose axis is vertical. In Case III. also we have positive precession. A good example of this occurs in the case of a coin spinning on a table when its plane is nearly horizontal.

107. Case II., that of a convex cone rolling inside a concave one, gives an example of negative precession: for when viewed as before on the hemispherical surface the direction of angular rotation of the instantaneous axis is opposite to that of the rolling cone. This is the case of a symmetrical cup (or figure of revolution) supported on a point, and stable when balanced, i.e., having its centre of gravity below the pivot; when inclined and set spinning non-nutationally. For instance, if a Troughton's top be placed on its pivot in any inclined position, and then spun off with very great angular velocity about its axis of figure, the nutation will be insensible; but there will be slow precession.

To this case also belongs the precessional motion of the earth's Model

axis; for which the

illustrating Precession of Equinoxes.

6

of the equi

noxes.

Precession ner edge of a fixed ring in space (directionally fixed, that is to say, but having the same translational motion as the earth's centre), and imagine a circular post or pivot of radius BI to be fixed to the earth with its centre at B. This ideal pivot rolling on the inner edge of the fixed ring travels once round the 52,240,000 feet-circumference in 25,868 years, and therefore its own circumference must be 5:53 feet. Hence BI=0.88 feet; and angle BOI, or B, = 0"-00867.

Free rotation of a

cally symmetrical about an

axis.

108. Very interesting examples of Cases I. and III. are furbody kineti- nished by projectiles of different forms rotating about any axis. Thus the gyrations of an oval body or a rod or bar flung into the air belong to Class I. (the body having one axis of less moment of inertia than the other two, equal); and the seemingly irregular evolutions of an ill-thrown quoit belong to Class III. (the quoit having one axis of greater moment of inertia than the other two, which are equal). Case III. has therefore the following very interesting and important application.

If by a geological convulsion (or by the transference of a few million tons of matter from one part of the world to another) the earth's instantaneous axis OI (diagram III., § 105) were at any time brought to non-coincidence with its principal axis of least moment of inertia, which (§§ 825, 285) is an axis of approximate kinetic symmetry, the instantaneous axis will, and the fixed axis OA will, relatively to the solid, travel round the solid's axis of greatest moment of inertia in a period of about 306 days [this number being the reciprocal of the most probable C- A value of (§ 828)]; and the motion is represented by the

diagram of Case III. with BI=306 × AI. Thus in a very little

1

less than a day (less by when BOI is a small angle)

306

I revolves round A. It is OA, as has been remarked by Maxwell, that is found as the direction of the celestial pole by observations of the meridional zenith distances of stars, and this line being the resultant axis of the earth's moment of

tion of a

momentum (§ 267), would remain invariable in space did no Free rota external influence such as that of the moon and sun disturb the body kineti cally symearth's rotation. When we neglect precession and nutation, metrical the polar distances of the stars are constant notwithstanding axis. the ideal motion of the fixed axis which we are now considering; and the effect of this motion will be to make a periodic variation of the latitude of every place on the earth's surface having for range on each side of its mean value the angle BOA, and for its period 306 days or thereabouts. Maxwell examined a four years series of Greenwich observations of Polaris (1851-2-3-4), and concluded that there was during those years no variation exceeding half a second of angle on each side of mean range, but that the evidence did not disprove a variation of that amount, but on the contrary gave a very slight indication of a minimum latitude of Greenwich belonging to the set of months Mar. '51, Feb. '52, Dec. '52, Nov. '53, Sept. '54.

“This result, however, is to be regarded as very doubtful.............. "and more observations would be required to establish the "existence of so small a variation at all.

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"I therefore conclude that the earth has been for a long time revolving about an axis very near to the axis of figure, if not "coinciding with it. The cause of this near coincidence is "either the original softness of the earth, or the present fluidity "of its interior [or the existence of water on its surface]. The axes of the earth are so nearly equal that a con"siderable elevation of a tract of country might produce a "deviation of the principal axis within the limits of observa"tion, and the only cause which would restore the uniform "motion, would be the action of a fluid which would gradually "diminish the oscillations of latitude. The permanence of "latitude essentially depends on the inequality of the earth's axes, for if they had all been equal, any alteration in the crust of the earth would have produced new principal axes, "and the axis of rotation would travel about those axes, alter

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* On a Dynamical Top, Trans. R. S. E., 1857, p. 559.