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.

XIX. The Propagation of Waves of Elastic Displacement along a Helical
Wire. By A. E. H. LOVE, M.A., F.R.S., Sedleian Professor of Natural
Philosophy in the University of Oxford.
[Received 4 December 1899.]
1. IT is known that the modes of vibration of an elastic wire or rod which in
the natural state is devoid of twist and has its elastic central line in the form of a
plane curve fall into two classes: in the first class the displacement is in the plane
of the wire and there is no twist; in the second class the displacement is at rightangles to the plane of the wire and is accompanied by twist. In particular for a
naturally circular wire forming a complete circle when the section of the wire is circular
and the material isotropic there are two modes of vibration with n wave-lengths to
the circumference; these belong to the first and second of the above classes respectively,
and their frequencies (p/27r) are given by the equations
_1 Ec2 n2 (n - 1)
l 4 poa4 1 + n2 '
~~~.and z-2 ~1 Ec2 12 (n2- 1)2
P 4 poa4 1 + Y + 2,
where a is the radius of the circle formed by the wire, c the radius of the section, p, the
mass per unit of length, E the Young's modulus and îî the Poisson's ratio of the material.
These results may be interpreted as giving the velocities with which two types of waves
travel round the circle.
So far little or nothing- appears to be known about the modes of vibration of wires of
which the central line in the natural state forrns a curve of double curvature, except that
the vibrations do not obviously fall into two classes related to the osculating plane in the:same way as the two classes for a plane curve are related to the plane of the curve.
The equation connecting the frequency with the wave-length when waves of elastic displacement are propagated along the wire has not been obtained; and although this
equation would obviously be quadratic when rotatory inertia is neglected, and so would
give two velocities of propagation for waves of a given length, it is by no means obvious
what would be the distinguishing marks of the two kinds of waves with the same wavelength.