each side being in fact equal to unity. At a point on the polhode cone x : y : z = p : q : r, and the equation of this cone is therefore
| A2 ( 1 − | Γ2 | ) x2 + B2 ( 1 − | Γ2 | ) y2 + C2 ( 1 − | Γ2 | ) z2 = 0. |
| 2AT | 2BT | 2CT |
(4)
Since 2AT − Γ2 = B (A − B)q2 + C(A − C)r2, it appears that if A > B > C the coefficient of x2 in (4) is positive, that of z2 is negative, whilst that of y2 is positive or negative according as 2BT ≷ Γ2. Hence the polhode cone surrounds the axis of greatest or least moment according as 2BT ≷ Γ2. In the critical case of 2BT = Γ2 it breaks up into two planes through the axis of mean moment (Oy). The herpolhode curve in the fixed plane is obviously confined between two concentric circles which it alternately touches; it is not in general a re-entrant curve. It has been shown by De Sparre that, owing to the limitation imposed on the possible forms of the momental ellipsoid by the relation B + C > A, the curve has no points of inflexion. The invariable line OH describes another cone in the 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 |
(5)
| 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)