(8)
Hence (5) will be satisfied, provided
| −σp0 | = | B − C | , | σq0 | = | C − A | , | −k2σr0 | = | A − B | . |
| q0r0 | A | r0p0 | B | p0q0 | C |
(9)
These equations, together with the arbitrary initial values of p, q, r, determine the six constants which we have denoted by p0, q0, r0, k2, σ, ε. We will suppose that A > B > C. From the form of the polhode curves referred to in § 19 it appears that the angular velocity q about the axis of mean moment must vanish periodically. If we adopt one of these epochs as the origin of t, we have ε = 0, and p0, r0 will become identical with the initial values of p, r. The conditions (9) then lead to
| q02 = | A (A − C) | p02, σ2 = | (A − C) (B − C) | r02, k2 = | A (A − B) | · | p02 | . |
| B (B − C) | AB | C (B − C) | r02 |
(10)
For a real solution we must have k2 < 1, which is equivalent to 2BT > Γ2. If the initial conditions are such as to make 2BT < Γ2, we must interchange the forms of p and r in (7). In the present case the instantaneous axis returns to its initial position in the body whenever φ increases by 2π, i.e. whenever t increases by 4K/σ, when K is the “complete” elliptic integral of the first kind with respect to the modulus k.
The elliptic functions degenerate into simpler forms when k2 = 0 or k2 = 1. The former case arises when two of the principal moments are equal; this has been sufficiently dealt with in § 19. If k2 = 1, we must have 2BT = Γ2. We have seen that the alternative 2BT ≷ Γ2 determines whether the polhode cone surrounds the principal axis of least or greatest moment. The case of 2BT = Γ2, exactly, is therefore a critical case; it may be shown that the instantaneous axis either coincides permanently with the axis of mean moment or approaches it asymptotically.