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.

366 PROF. LOVE, THE PROPAGATION OF WAVES
The fact that there are two waves with different velocities suggests an analogy
with the optical theory of rotatory polarization, and leads to the question whether in
any sense the two waves can be regarded as right-handed and left-handed. The most
obvious possibility of this kind would be that /3 and w should be always in the saine
phase for one wave and in opposite phases for the other; it is found however that
this is not the case; another possibility would be that the component displacements
parallel to the axis and to the circulai section of the cylinder on which the helix is
traced should be everywhere directed like a right-handed system of axial and circular
translatory displacements for one wave and like a corresponding left-handed system for
the other; this also is found not to be the case. It appears that up to the degree
of approximation which is usually included in the theory of elastic wires there is no
rotatory effect involved.
In three particular cases it is found that the equation for the frequency of waves of
given length breaks up into two separate equations. This happens (a) when n2=l/p2+l/o-2,
(b) when m2 = l/p - /o2, (c) when the helix is very fiat or 1/o can be neglected. In
case (a) one of the modes of deformnation is equivalent to a rigid body displacement
of the helix at right angles to its axis, and the corresponding speed of course vanishes;
in case (c) the types correspond to the two already known for a circle; in case (b) the two
types are distinguished by the vanishing of the flexural couples in and perpendicular
to the osculating plane; this case occurs only if the angle of the helix is less than 47r.
3. The wire is taken to be of uniform circular section (radius c), and of homogeneous isotropic material, and in the natural state the line of centres of its sections
forms a circular helix of curvature l/p and tortuosity 1/o. The displacement of a point
on the central line is specified by components u, v, w along the principal normal, the
binormal a.nd the tangent in the senses already defined, but it is necessary to fix the
meaning of the angular displacement fi. For this purpose we suppose a frame of three
coorthogonal lines to move along the helix so that the three lines always coincide
with the principal normal, the binormal, and the tangent; if the origin of the frame
moves with unit velocity the lines of the frame will rotate with an angular velocity
which has components i/p about the binormal and 1I/o about the tangent. We can
construct a corresponding frame for the strained wire by taking as origin the displaced
position of a point on the strained elastic central line, as one line of reference the
tangent to the strained elastic central line through the point, and as one plane of
reference the plane through this line which contains the tangent to that line of particles
which in the natural state coincided with the principal normal; when the displacement
is everywhere very small the lines of this frame very nearly coincide with those of the
firame attached to the unstrained wir,, and the plane of reference just defined makes
a very small angle with the osculating plane of the helix at the corresponding point; this
angle is /8. The "twist" of the wire is expressed by
1 h ci e 1 eIv ou \
-+ - - l +
o as p \as o
where cls is the element of arc of the helix.