provided x1 : y1 : z1 be the corresponding values of the ratios x:y:z. Again, the expression (22) will also have a least value when the ratios x : y : z are subject to the condition
| x1 | ∂V | + y1 | ∂V | + z1 | ∂V | = 0; |
| ∂x | ∂y | ∂z |
(24)
and if this be denoted by σ22 we have a second system of equations similar to (23). The remaining value σ22 is the value of (22) when x : y : z arc chosen so as to satisfy (24) and
| x2 | ∂V | + y2 | ∂V | + z2 | ∂V | = 0; |
| ∂x | ∂y | ∂z |
(25)
The problem is identical with that of finding the common conjugate diameters of the ellipsoids T(x, y, z) = const., V(x, y, z) = const. If in (21) we imagine that x, y, z denote infinitesimal rotations of a solid free to turn about a fixed point in a given field of force, it appears that the three normal modes consist each of a rotation about one of the three diameters aforesaid, and that the values of σ are proportional to the ratios of the lengths of corresponding diameters of the two quadrics.
We proceed to the forced vibrations of the system. The typical case is where the extraneous forces are of the simple-harmonic type cos (σt + ε); the most general law of variation with time can be derived from this by superposition, in virtue of Fourier’s theorem. Analytically, it is convenient to put Qr, equal to eiσt multiplied by a complex coefficient; owing to the linearity of the equations the factor eiσt will run through them all, and need not always be exhibited. For a system of one degree of freedom we have
aq̈ + cq = Q,
(26)