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. 369
6. We may now substitute for gq and g, in terrns of V and W and obtain by
elimination of V and W the equation for p2
IÀ + 1m + A1 < _+ )
( A )(~ i2p2)( CM2p2 A [(2 +s2p2 +C?,2p2) + p2 2
+ m2 + - +) -(i- (i C2p = ~o.........(13)
If m2n- - - does not vanish this equation can be written
p2
{1 -( + o) } -(1 + 1) X2 (1 - /- - K2)2 8K (2 + K- K1-)=O............(14),
by putting
/CK=A/Cm2p2, ic, = l/m2p, = 1/,n2o-2)
x = A - (1 - -2 p p2 -4..........................
Since A/C =1 + 7, where?v is the Poisson's ratio of the material,
CK0 - K1 = 77K,
and this is always positive; so that, if for x in the left-hand member of the above
equation (14) we substitute the values oo, l/(l+eK), 1/(1 + K), 0, the expression has
the signs + - -+, and thus one of the two values of x exceeds l/(l + i) and the
other is less than 1/(1 + - o), both values being positive. It follows that there are two
possible velocities for waves of given length, the speed of one exceeding
r A mp2 (qn2- _ 1/p2 _ 1/U2)2 2
Lpoo 1 + m2p J
and that of the other being less than
A m2p2 (m2 - i/p - l/o-2)22]
Lpo ~1+,+fm2p j'
these two expressions become the speeds of the corresponding waves round a circular
ring by writing n for mp and omitting 1/o-.
The left-hand member of the equation (14) for x breaks up into factors rational
in Ko, K1, K2 if
[(2 + Ko + K1) (1 - l - fK)2 + SK2 (c2 + K - K)] - 4 (1 + K0) (1 + K1) (1 - KC1- -K_4
is the square of a rational function of K0, K1, K,. This is the case when_.2 + K/ - K = 0,
or when 1- Kc - K = 0, or when K = 0, or when 1- I + 2, = O, for in the last case the
expression becomes
16 (1 - K,)2 [(1-,) (KC- )- 2 (K + 1)]2.
Of these cases the first cannot happen since Ko > Ki, and the third is the limiting
case in which the helix becomes a circle; the two remaining cases will be discussed
later.
VOL. XVIII. 47