da] be less than the circumference; but unstable if [lambda] be
greater than the circumference of the cylinder. Disturbances of the
former kind lead to _vibrations_ of harmonic type, whose amplitudes
always remain small; but disturbances, whose wave-length exceeds the
circumference, result in a greater and greater departure from the
cylindrical figure. The analytical expression for the motion in the
latter case involves exponential terms, one of which (except in case of
a particular relation between the initial displacements and velocities)
increases rapidly, being equally multiplied in equal times. The
coefficient (q) of the time in the exponential term (e^{qt}) may be
considered to measure the degree of dynamical instability; its
reciprocal 1/q is the time in which the disturbance is multiplied in the
ratio 1 : e.

The degree of instability, as measured by q, is not to be determined
from statical considerations only; otherwise there would be no limit to
the increasing efficiency of the longer wave-lengths. The joint
operation of superficial tension and _inertia_ in fixing the wave-length
of maximum instability was first considered by Lord Rayleigh in a paper
(_Math. Soc. Proc._, November 1878) on the "Instability of Jets." It
appears that the value of q may be expressed in the form

/ / T \
q = / ( -------- ).F(ka), (2)
\/ \[rho]a^3/

where, as before, T is the superficial tension, [rho] the density, and F
is given by the following table: --

+--------+--------+--------+--------+
| k^2a^2.| F(ka). | k^2a^2.| F(ka). |
+--------+--------+--------+--------+
| .05 | .1536 | .4 | .3382 |
| .1 | .2108 | .5 | .3432 |
| .2 | .2794 | .6 | .3344 |
| .3 | .3182 | .8 | .2701 |
| | | .9 | .2015 |
+--------+--------+--------+--------+

The greatest value of F thus corresponds, not to a zero value of k^2a^2,
but approximately to k^2a^2 = .4858, or to [lambda] = 4.508 X 2a. Hence
the maximum instability occurs when the wave-length of disturbance is
about half as great again as that at which instability first commences.

Taking for water, in C.G.S. units, T = 81, [rho] = 1, we get for the
case of maximum instability

a^(3/2)
q_(-1) = --------- = .115d^(3/2) (3),
81 X .343

if d be the diameter of the cylinder. Thus, if d = 1, q^(-1) = .115; or
for a diameter of one centimetre the d