to, of
which only the title has hitherto been published, I showed that, from the
mathematical investigation of a gyrostatically dominated combination
contained in the passage of Thomson and Tait's "Natural Philosophy"
referred to, it follows that any ideal system of material particles, acting
on one another mutually through massless connecting springs, may be
perfectly imitated in a model consisting of rigid links jointed together,
and having rapidly rotating fly wheels pivoted on some or on all of the
links. The imitation is not confined to cases of equilibrium. It holds also
for vibration produced by disturbing the system infinitesimally from a
position of stable equilibrium and leaving it to itself. Thus we may make a
gyrostatic system such that it is in equilibrium under the influence of
certain positive forces applied to different points of this system; all the
forces being precisely the same as, and the points of application similarly
situated to, those of the stable system with springs. Then, provided proper
masses (that is to say, proper amounts and distributions of inertia) be
attributed to the links, we may remove the external forces from each
system, and the consequent vibration of the points of application of the
forces will be identical. Or we may act upon the systems of material points
and springs with any given forces for any given time, and leave it to
itself, and do the same thing for the gyrostatic system; the consequent
motion will be the same in the two cases. If in the one case the springs
are made more and more stiff, and in the other case the angular velocities
of the fly wheels are made greater and greater, the periods of the
vibrational constituents of the motion will become shorter and shorter, and
the amplitudes smaller and smaller, and the motions will approach more and
more nearly those of two perfectly rigid groups of material points moving
through space and rotating according to the well known mode of rotation of
a rigid body having unequal moments of inertia about its three principal
axes. In one case the ideal nearly rigid connection between the particles
is produced by massless, exceedingly stiff springs; in the other case it is
produced by the exceedingly rapid rotation of the fly wheels in a system
which, when the fly wheels are deprived of their rotation, is perfectly
limp.

[Footnote 1: Paper on "Vortex Atoms," _Proc_. R.S.E. February. 1867:
abstract of a lecture before the Royal Ins