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. 371
V/ W, but in the present case V and W are not independent. The equation connecting
them is obtained by eliminating x from (17) and (20) in the form
(Ko-K) (mr2 — 2)%= (2 +Â/pî) {4m2 (i * p2) [1+ -Po (m2 - + )
2 = 12_ 1V + 1 + 2p( 1 )
Po- W p2 0 olCr+p2J
or 2 {V2- W2 ( + 2p2)} (m2- 2 +
-wv -P 2m;p2 m /C — (m =......(22),
which gives two values for V/W, having a negative product, and thus showing that
in the two waves the values of V/W have opposite signs. We now substitute for V/W
the values
+~o0, o, p m2 _+, _ 2_,m2_ 1+ 1,
placing these values in order of decreasing algebraic magnitude. For shortness we write
- (m2 1 + 1
2 2
-P+a......................... (23),
(n7 p2) p2 1/2) J
and then a >/ according as (m2-l/p2+1/o-2)<0. There are three cases depending on
the signs of 2 -l /p2 and of m2 _ l/p2 + 1/o2. In any case when we substitute V/ W =,
the left-hand inember of (22) becomes
p_2_ _ 2 \2 2(
P22l2 r -a2p + A!C - 1 - v p-2 - 1 + o 2
and when we substitute V/W==/ the left-hand member of (22) becomes
-4W2 Mo2( +A/C) ( 1 /,2 _ 1 1
~*" 2 2Y+AIC- 1 P2 0'/ p/ 2 1\
Now in the slower wave we have
W
1-a-< 0,
which shows that if m2' - l/p2+ 1/o2 is positive
0 > V/W>a,
and if m2 - 1/p2 + 1/o- is negative
a> V/W > 0.
In the quicker wave we have
(w2,2 - /p2 + 1/o-2) (-Ç- f) >0.
47-2