Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

PROF. LOVE, THE PROPAGATION OF WAVES, ETC. 365
It seemed to me that it would be not without interest to seek to answer the
questions thus proposed in the case of a wire which in the natural state has its elastic
central line in the form of a helix. As regards the free vibrations of a terminated portion
of such a wire with free ends, or fixed ends, or under the action of given forces at
the terminals, it would be possible to form the equation for the frequency, but the
equation appears to be so complicated as to be quite uninterpretable; and in fact in
the simpler problem presented by a circular wire with ends, which has been treated
in some detail by Lamb*, it appears that to interpret the results the total curvature
must be taken to be slight, and the results which can then be obtained are such as
might be reached by suitable approximate methods. In the case of a helical wire the
most important of all the problems of vibration is that of a spiral spring supporting
a weight which oscillates up and down; and this can be treated adequately by means
of an approximate theory in which the wire is taken to have at any time the form
of the helix corresponding to its axial length and to the position of the load. The
problem of the propagation of waves along an infinite helical wire remains. I have
found that in general for a given wave-length two types of waves are propagated with
different velocities; in both types all the kinds of displacement (tangential, normal and
torsional) are involved, and there is no rational relation between the different displacements
which serves to distinguish the types of the two waves, but these types are finally
and completely separated by a circumstance of phase in the different components of
the displacement.
2. The helix which is the natural form of the elastic central line of the wire
may be thought of as traced on a circular cylinder, and then any particle on this line
undergoes a displacement which may be resolved into components u, v, w along the
principal normal, the binormal and the tangent to the helix. The principal normal
coincides with the radius of the cylinder, and the displacement u is reckoned positive
when it is inwards along this normal; the displacement w is reckoned positive when
it is in the sense in which the arc is measured, and then the positive sense of the
displacement v is determined by the convention that the
positive directions of u, v, w are a right-handed system
for a right-handed helix. Further there is an angular
displacement by rotation of the sections, of amount 3,
about the tangent to the helix, and /8 is reckoned positive /
when /3 and w form a right-handed rotation and translatory displacement. Now it is found that in general the \
two waves of given length that can be propagated are
distinguished according as the displacements v and w are 1
in the same phase or in opposite phases at all points of
the helix. If 1/p and 1/o- are the measures of curvature
and tortuosity of the helix, and 27r/mn is the wave-length, then in the quicker wave v
and w are everywhere in the same phase, and in the slower wave they are in opposite
phases, provided mn2 > l/p2 - I/or2, but if m2 < 1/p2 - 1/ao2 this relation is reversed.
' Poc. Lo1nd. Math. Soc., xIx. 1888.