where a1r, a2r, ... anr are the minors of the rth row of the determinant (7). Every particle of the system executes in general a simple vibration of the imposed period 2π/σ, and all the particles pass simultaneously through their equilibrium positions. The amplitude becomes very great when σ2 approximates to a root of (6), i.e. when the imposed period nearly coincides with one of the free periods. Since ars = asr, the coefficient of Qs in the expression for qr is identical with that of Qr in the expression for qs. Various important “reciprocal theorems” formulated by H. Helmholtz and Lord Rayleigh are founded on this relation. Free vibrations must of course be superposed on the forced vibrations given by (29) in order to obtain the complete solution of the dynamical equations.
In practice the vibrations of a system are more or less affected by dissipative forces. In order to obtain at all events a qualitative representation of these it is usual to introduce into the equations frictional terms proportional to the velocities. Thus in the case of one degree of freedom we have, in place of (26),
aq̈ + bq̇ + cq = Q,
(30)
where a, b, c are positive. The solution of this has been sufficiently discussed in § 12. In the case of multiple freedom, the equations of small motion when modified by the introduction of terms proportional to the velocities are of the type
| d | ∂T | + B1rq̇1 + B2rq̇2 + ... + Bnrq̇n + | ∂V | = Qr. | |
| dt | ∂q̇r | ∂qr |
(31)
If we put
brs = bsr = 1⁄2 (Brs + Bsr), βrs = −βsr = 1⁄2 (Brs − Bsr),
(32)