398 THE THEORY OF SCREWS. [359
<f> which is evoked by the action of the forces. It will, of course, generally
happen that the chain 0 is different from the chain </>. It can however be
shown that 0 and </> are not in every case distinct. There are n different
screw-chains, each of which regarded as a will have the two corresponding
screws 0 and </> identical. Nor is it difficult to see what the effect of such a
displacement must be on the small oscillations which follow. A wrench is
evoked by the displacement, and since 0 and <f> coincide, that wrench is
undistinguishable from an infinitely small impulsive wrench which will
make the system commence to twist about a. We are thus led to the
result that—
There are n screw-chains such that if the system be displaced by a twist
about one of these screw-chains, and then released, it will continue for ever to
twist to and fro on the same screw-chain.
Following the language previously used, we may speak of these as
harmonic screw-chains, and it can be shown that whatever be the small
displacement of the system, and whatever be the small initial velocities with
which it is started, the small oscillations are merely compounded of twist
vibrations about the n harmonic screw-chains.