body, called the invariable cone. At any point of this we have
x : y : z = Ap · Bq : Cr, and the equation is therefore
| 1 − | Γ2 | x2 + 1 − | Γ2 | y2 + 1 − | Γ2 | z2 = 0. |
| 2AT | 2BT | 2CT |
 |
| Fig. 80. |
The signs of the coefficients follow the same rule as in the case of (4). The possible forms of the invariable cone are indicated in fig. 80 by means of the intersections with a concentric spherical surface. In the critical case of 2BT = Γ2 the cone degenerates into two planes. It appears that if the body be sightly disturbed from a state of rotation about the principal axis of greatest or least moment, the invariable cone will closely surround this axis, which will therefore never deviate far from the invariable line. If, on the other hand, the body be slightly disturbed from a state of rotation about the mean axis a wide deviation will take place. Hence a rotation about the axis of greatest or least moment is reckoned as stable, a rotation about the mean axis as unstable. The question is greatly simplified when two of the principal moments are equal, say A = B. The polhode and herpolhode cones are then right circular, and the motion is “precessional” according to the definition of § 18. If α be the inclination of the instantaneous axis to the axis of symmetry, β the inclination of the latter axis to the invariable line, we have
| Γ cos β = C ω cos α, Γ sin β = A ω sin α, | (6) |
whence
| tan β = AC tan α. | (7) |
 |
| Fig. 81. |
Hence β ≷ α, and the circumstances are therefore those of the first or second case in fig. 78, according as A ≷ C. If ψ be the rate at which the plane HOJ revolves about OH, we have
| ψ = | sin α | ω = | C cos α | ω, |
| sin β | A cos β |
by § 18 (3). Also if χ. be the rate at which J describes the
polhode, we have ψ. sin (β − α) = χ. sin β, whence
| χ. = | sin (α − β) | ω. |
| sin α |
If the instantaneous axis only deviate slightly from the axis of
symmetry the angles α, β are small, and χ. = (A − C) A·ω; the
instantaneous axis therefore completes its revolution in the body
in the period
| 2π | = | A − C | ω. |
| χ. | A |
In the case of the earth it is inferred from the independent phenomenon of luni-solar precession that (C − A)/A = .00313. Hence if the earth’s axis of rotation deviates slightly from the axis of figure, it should describe a cone about the latter in 320 sidereal days. This would cause a periodic variation in the latitude of any place on the earth’s surface, as determined by astronomical methods. There appears to be evidence of a slight periodic variation of latitude, but the period would seem to be about fourteen months. The discrepancy is attributed to a defect of rigidity in the earth. The phenomenon is known as the Eulerian nutation, since it is supposed to come under the free rotations first discussed by Euler.
§ 20. Motion of a Solid of Revolution.—In the case of a solid of revolution, or (more generally) whenever there is kinetic symmetry about an axis through the mass-centre, or through a fixed point O, a number of interesting problems can be treated almost directly from first principles. It frequently happens that the extraneous forces have zero moment about the axis of symmetry, as e.g. in the case of the flywheel of a gyroscope if we neglect the friction at the bearings. The angular velocity (r) about this axis is then constant. For we have seen that r is constant when there are no extraneous forces; and r is evidently not affected by an instantaneous impulse which leaves the angular momentum Cr, about the axis of symmetry, unaltered. And a continuous force may be regarded as the limit of a succession of infinitesimal instantaneous impulses.
 |
| Fig. 82. |
Suppose, for example, that a flywheel is rotating with angular velocity n about its axis, which is (say) horizontal, and that this axis is made to rotate with the angular velocity ψ. in the horizontal plane. The components of angular momentum about the axis of the flywheel and about the vertical will be Cn and A ψ. respectively, where A is the moment of inertia about any axis through the mass-centre (or through the fixed point O) perpendicular to that of symmetry. If OK→ be the vector representing the former component at time t, the vector which represents it at time t + δt will be OK→′, equal to OK→ in magnitude and making with it an angle δψ. Hence KK→′ (= Cn δψ) will represent the change in this component due to the extraneous forces. Hence, so far as this component is concerned, the extraneous forces must supply a couple of moment Cnψ. in a vertical plane through the axis of the flywheel. If this couple be absent, the axis will be tilted out of the horizontal plane in such a sense that the direction of the spin n approximates to that of the azimuthal rotation ψ. . The remaining constituent of the extraneous forces is a couple Aψ.. about the vertical; this vanishes if ψ. is constant. If the axis of the flywheel make an angle θ with the vertical, it is seen in like manner that the required couple in the vertical plane through the axis is Cn sin θ ψ. . This matter can be strikingly illustrated with an ordinary gyroscope, e.g. by making the larger movable ring in fig. 37 rotate about its vertical diameter.
 |
| Fig. 83. |
If the direction of the axis of kinetic symmetry be specified by means of the angular co-ordinates θ, ψ of § 7, then considering the component velocities of the point C in fig. 83, which are θ. and sin θψ. along and perpendicular to the meridian ZC, we see that the component angular velocities about the lines OA′, OB′ are −sin θ ψ. and θ. respectively. Hence if the principal moments of inertia at O be A, A, C, and if n be the constant angular velocity about the axis OC, the kinetic energy is given by
Again, the components of angular momentum about OC, OA′ are Cn, −A sin θ ψ. , and therefore the angular momentum (μ, say) about OZ is
We can hence deduce the condition of steady precessional motion in a top. A solid of revolution is supposed to be free to turn about a fixed point O on its axis of symmetry, its mass-centre G being in this axis at a distance h from O. In fig. 83 OZ is supposed to be vertical, and OC is the axis of the solid drawn in the direction OG. If θ is constant the points C, A′ will in time δt come to positions C″, A″ such that CC″ = sin θ δψ, A′A″ = cos θ δψ, and the angular momentum about OB′ will become Cn sin θ δψ − A sin θ ψ. · cos θ δψ. Equating this to Mgh sin θ δt, and dividing out by sin θ, we obtain
as the condition in question. For given values of n and θ we have two possible values of ψ. provided n exceed a certain limit. With a very rapid spin, or (more precisely) with Cn large in comparison with √(4AMgh cos θ), one value of ψ. is small and the other large, viz. the two values are Mgh/Cn and Cn/A cos θ approximately. The absence of g from the latter expression indicates that the circumstances of the rapid precession are very
