(12)
The roots of this quadratic in σ2 are easily seen to be real and positive. If M be large compared with m, μ is small, and the roots are g/a and g/b, approximately. In the normal mode corresponding to the former root, M swings almost like the bob of a simple pendulum of length a, being comparatively uninfluenced by the presence of m, whilst m executes a “forced” vibration (§ 12) of the corresponding period. In the second mode, M is nearly at rest [as appears from the second of equations (11)], whilst m swings almost like the bob of a simple pendulum of length b. Whatever the ratio M/m, the two values of σ2 can never be exactly equal, but they are approximately equal if a, b are nearly equal and μ is very small. A curious phenomenon is then to be observed; the motion of each particle, being made up (in general) of two superposed simple vibrations of nearly equal period, is seen to fluctuate greatly in extent, and if the amplitudes be equal we have periods of approximate rest, as in the case of “beats” in acoustics. The vibration then appears to be transferred alternately from m to M at regular intervals. If, on the other hand, M is small compared with m, μ is nearly equal to unity, and the roots of (12) are σ2 = g/(a + b) and σ2 = mg/M·(a + b)/ab, approximately. The former root makes θ = φ, nearly; in the corresponding normal mode m oscillates like the bob of a simple pendulum of length a + b. In the second mode aθ + bφ = 0, nearly, so that m is approximately at rest. The oscillation of M then resembles that of a particle at a distance a from one end of a string of length a + b fixed at the ends and subject to a tension mg.
The motion of the system consequent on arbitrary initial conditions may be obtained by superposition of the n normal modes with suitable amplitudes and phases. We have then
qr = αrθ + αr′θ′ + αr″θ″ + ...,
(13)
where
θ = C cos (σt + ε), θ′ = C′ cos (σ′t + ε), θ″ = C″ cos (σ″t + ε), ...
(14)
provided σ2, σ′2, σ″2, ... are the n roots of (6). The coefficients of θ, θ′, θ″, ... in (13) satisfy the conjugate or orthogonal relations
a11α1α1′ + a22α2α2′ + ... + a12 (α1α2′ + α2α1′) + ... = 0,