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, 367
4. The action of the part of the wire for which s is greater upon the part for
which s is less, across any section, can be reduced to a resultant force at the centre
of the section and a couple. The force mnay be resolved into components N1 along the
principal normal, V2 along the binormal, and T along the tangent, in the senses in which
u, v, 'w are reckoned positive. The couple may be resolved into two flexural couples G1, G2
and a torsional couple H about the same three lines. The couples are expressible in ternis
of the displacements by the equations
G =A -_ (+ -- 1-( _+ W]
ap as\(s (- as O- a P
G a a(u v +\ (a v \u
G,2=A Las-+ p)- a S + -)........................ (1),
as\as o-pi o-kas o]
H - CLafl a ) +-]
|a+ p as OJ -
in which A, = lErc4, is the flexural rigidity, and C, = Erc4/(1 + v), is the torsional
rigidity.
Further, the displacements u, w are connected by the relation of inextensibility of
the wire
azu _
as p -(2)
When rotatory inertia is neglected the stress-couples are connected with the stressresultants and with each other by the three equations of moments
aG, Go H
-+ -- - V2 = O,
as o- p
2 + G0,...(:3).
as o
aH G,
__ _ G= t)
as p
The equations of sinall motion are the three equations of resolution
811 NV 2 l \
ds- + p- = po0)o t
as o- p at
+ =PoW,l.................(4),
as ataT i aV,-w
a-s = po)O a
in which po is the density of the material of the wire and w, = rc2, is the area of the
cross-section.
5. We shall now suppose that simple harmonic waves are propagated along the wire,
and take as expressions for the displacements
= - mp W sin (ms -pt), v = Fcos (ms -p(t), w = W cos (ms -pt),); = B cos (ms- pt).