(22) − z(18) + x(20) = 0,
(30)
gives
| ( | A | + x2 + z2 ) | dq | − p | h3 | + ( | A | + z2 ) p2 cot θ + p2xz |
| M | dt | M | M |
+ q2ρ (x cos θ − z sin θ) − prx (x + z cot θ) − g (x cos θ − z sin θ) = 0,
(C)
and this combined with (A) and (B) will lead to an equation the integral of which is the equation of energy.
13. The equations (A) (B) (C) are intractable in this general form; but the restricted case may be considered when the axis moves in steady motion at a constant inclination α to the vertical; and the stability is secured if a small nutation of the axis can be superposed.
It is convenient to put p = Ω sin θ, so that Ω is the angular velocity of the plane Gzx about the vertical; (A) (B) (C) become