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.

372 PROF. LOVE, THE PROPAGATION OF WAVES
Thus when m2 - l/p2+l/o-2 and n2 - 1/p2 are both positive we have
V/W> /3,
when m2 - 1/p2 + 1/o2 is positive and m2- 1/p2 is negative
V/W> /3>0,
when m2 - l/p2 + I/o-2 and m2 - llp2 are both negative
O >/3 > / W.
When m2 - 1/p2 and m2 - 1/p2 + 1/o2 are both positive we obtain after substitution
in the left-hand member of (22) the signs shown in the table
V/W = oo 0 / a -oo
+ - -+ +
When m2 - 1/p2 is negative and m2 - l/p2 + l/o2 is positive we obtain
V/W = oo ( O a -o
+ - - + +
When m2- /p2 and m2 - 1/p2+ 1/o-2 are both negative we obtain
V/W = oo a O 8 -oo
-_ - + -
By comparison of these results we see that when m2 - 1/p2 + 1/o-2 is positive V/W
is positive in the quicker wave and negative in the slower one, but when mn2 - 1/p2 + 1/2
is negative the reverse is the case. When V/W is positive the displacements v and w are
in the same phase at all points of the helix, and when V/W is negative these displacements are everywhere in opposite phases.
8. If the helix of angle a is wound on a cylinder of radius a the displacement
parallel to the axis is a sec a (w/o- + v/p), and the displacement parallel to the circular
section is a sec a (w/p-v v/), and the wave is in a certain sense right-handed or lefthanded according as
(W/p -V/a) (W/l + V/p)
is positive or negative. We write e for this, and then the values of: in the two waves
satisfy the equation
2 [(1/p- /)2 - (1 + r2p2) (e/p + I/o)2] (m2 - /lp2 + 1/0o2)
(8 / [P A/C- 1 (2 1 1) 2
(p O- (p U )I- [p 9mnp2 +- 4/C -1 p2 -2).and the two waves will be respectively right-handed and left-handed if the roots have
*opposite signs. To show that this is not always the case it is sufficient to take rz
very great and substitute for: in the left-hand member the values
O - p/o- - o;
the signs are - + -,