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(12)

where Ci;, <o,, /,, K,, K, (/, j = 1,2, 3), /3, J, and cr are defined as follows: Let a, (/ = 1, 2, 3) be a dextral set of unit vectors parallel to the central principal axes of inertia of G, with a3 parallel to the axis of B (a,, a2, a3 are thus fixed in A ); and introduce unit vectors e2 and e3 (see Fig. 3.11.1) such that el5 e2, e3 form a dextral, orthogonal set with e2 — (Ndet/dt)D,~\ so that e3 is normal to the orbit plane; let

and take

Next, with

Finally, let /3, J, and o-have the same meaning as in Sec. 3.7. G can move in such a way that a, = e, (i = 1,2, 3) while AtoB • a3 has the constant value Aa>B. This motion is unstable if

or if any one of the following is satisfied:

b < 0 c < 0 b2 — c < 0 where b and c are given by iXK2 + 3K2+tA4V{hK>-hKi)(AûBV

I2il

Derivations Equations (4)—(9) are identical with Eqs. (3.5.4)-(3.5.9) respectively. For the derivation of the latter, see Eqs. (3.5.28)-(3.5.33).

If M, = M • a, (i = 1, 2, 3), where M is given by Eq. (2), then, making use of Eqs. (3), (16), and (2.6.8), one can write

M, = -3ÎÎ2/,/:,C12C13 M2 = -3il2I2K2CuCu M3 = -3Ù2I3K3CuC{2

and substitution into Eqs. (3.7.9)—(3.7.11) leads directly to equations that can be seen to be equivalent to Eqs. (10)—(12), once Eqs. (17) have been brought into play.

When G moves in such a way that a, = e, (/ = 1, 2, 3), then and

C M — 1 — C i2 — C,3 — C3, — C32 — C33 —1—0

To verify that this is a possible motion, note that Eqs. (4)—(12) are satisfied when Cjj and (Oj (i = 1, 3\j = 1, 2, 3) have values compatible with Eqs. (25) and (26); and, to establish conditions under which the motion is unstable, introduce perturbations Cjj and dij (i = l,3;y = 1, 2, 3) as in Eqs. (3.5.71)—(3.5.73), substitute into Eqs. (4)—(12), and linearize the resulting equations, which produces t C3i = C33 fl - (o2 C32 = 2>, - C3t il C33 = 0 (27)

o>3 = -3ft2*:3c,2 Also, note that [see Eqs. (1.2.14) with / = l,j = 3]

so that, in view of Eqs. (3.5.71) and (3.5.72), one has after linearization, c3, + C,3 = 0 (33)

Now solve the second of Eqs. (28) for 2J3, substitute into Eq. (31), and use the third of Eqs. (27) to obtain

and refer to the first two of Eqs. (27) and (28) together with Eq. (33) to write

C3I

0 0