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.

OF ELASTIC DISPLACEMENT ALONG A HELICAL WIRE. 373
showing that both values of e are negative, and both waves are left-handed in this
sense when m is sufficiently great.
A similar method may be applied to show that there are values of m for which
both values of /3/w have the same sign, and thus the waves are not respectively righthanded and left-handed in regard to 3 and w.
9. We have already noted that in three special cases the equation (13) for p2 can be
solved rationally in terms of m, p, a.
Taking n2 = 1/p2 + 1/o2 = a-2 cos2 a, it is convenient to put ms= 0, and then 0 is the
angle turned through by the radius of the helix about the axis of the cylinder in
passing along the curve from the point frorn which the arc is measured to the point
at which the arc is s.
In this case equations (12) become
( l \ 2P poo2 (1
( 1+ g2- ) ~AP (l+m2p2)W,
(g + g2m) - -PP V,
and equations (8) and (9) become
gl (+ Cmp = V+ W,
g2-(V
so that mY2 + gl/o = { 2 + (A} v+ Tr P)
and thus either V+ Wp/o = O and p=, or else
V= W (1 + m2p2),
P
ans Po~~P2(l +m2p) 4m(2 + )/ (2 + * +p)
The second kind of motion is an example of the quicker wave, and the speed p
is given by,2 8A (p2 + o2)2 (A + ()) 2 + 2Cp2 1
~ p0o p2a4 (A + C) 2 + Cp2 (22 + p2)
In the displacement for which p=0 the equation V+ Wp/- =O shows that there
is no displacement parallel to the axis of the helix; we also have
W cos a - V sin a = W (cos a + sin a tan a) = W sec a,