Eliminating the ratio A : B we have
(σ2 + ω2 − p2) (σ2 + ω2 − q2) − 4σ2ω2 = 0.
(38)
It is easily proved that the roots of this quadratic in σ2 are always real, and that they are moreover both positive unless ω2 lies between p2 and q2. The ratio B/A is determined in each case by either of the equations (37); hence each root of the quadratic gives a solution of the type (36), with two arbitrary constants A, ε. Since the equations (35) are linear, these two solutions are to be superposed. If the quadratic (38) has a negative root, the trigonometrical functions in (36) are to be replaced by real exponentials, and the position x = 0, y = 0 is unstable. This occurs only when the period (2π/ω) of revolution of the arm lies between the two periods (2π/p, 2π/q) of oscillation when the arm is fixed.
§ 14. Central Forces. Hodograph.—The motion of a particle subject to a force which passes always through a fixed point O is necessarily in a plane orbit. For its investigation we require two equations; these may be obtained in a variety of forms.
Since the impulse of the force in any element of time δt has zero moment about O, the same will be true of the additional momentum generated. Hence the moment of the momentum (considered as a localized vector) about O will be constant. In symbols, if v be the velocity and p the perpendicular from O to the tangent to the path,
pv = h,
(1)
where h is a constant. If δs be an element of the path, pδs is twice the area enclosed by δs and the radii drawn to its extremities from O. Hence if δA be this area, we have δA = 1⁄2 pδs = 1⁄2 hδt, or
| dA | = 1⁄2h |
| dt |