434 ON CONVECTION CURRENTS IN A [412
in any particular region depends upon the initial circumstances in and near that region, and these are supposed to be matters of chance*. The superposition of infinite trains of waves whose wave-lengths cluster round a given value raises the same questions as we are concerned with in considering the character of approximately homogeneous light.
In the present problem the case is much more complicated, unless we arbitrarily limit it to two dimensions. The cells of Benard are then reduced to infinitely long strips, and when there is instability we may ask for what wave-length (width of strip) the instability is greatest. The answer can be given under certain restrictions, and the manner in which equilibrium breaks down is then approximately determined. So long as the two-dimensional character is retained, there seems to be no reason to expect the wave-length to alter afterwards. But even if we assume a natural disposition to a two-dimensional motion, the direction of the length of the cells as well as the phase could only be determined by initial circumstances, and could not be expected to be uniform over the whole of the infinite plane.
According to the observations of Benard, something of this sort actually occurs when the layer of liquid has a general motion in its own plane at the moment when instability commences, the length of the cellular strips being parallel to the general velocity. But a little later, when the general motion has decayed, division-lines running in the perpendicular direction present themselves.
In general, it is easy to recognize that the question is much more complex. By Fourier's theorem the motion in, its earlier stages may be analysed into components, each of which corresponds to rectangular cells whose sides are parallel to fixed axes arbitrarily chosen. The solution for maximum instability yields one relation between the sides of the rectangle, but no indication of their ratio. It covers the two-dimensional case of infinitely long rectangles already referred to, and the contrasted case of squares for which the length of the side is thus determined. I do not see that any plausible hypothesis as to the origin of the initial disturbances leads us to expect one particular ratio of sides in preference to another.
On a more general view it appears that the function expressing the disturbance which develops most rapidly may be assimilated to that which represents the free vibration of an infinite stretched membrane vibrating with given frequency.
The calculations which follow are based upon equations given by Bous-sinesq, who has applied them to one or two particular problems. The special limitation which characterizes them is the neglect of variations of density,
* When a jet of liquid is acted on by an external vibrator, the resolution into drops may be regularized in a much higher degree.. Math. Soc. Yol. x. p. 4 (1879); Scientific Papers, Yol. i. p. 361. Also Theory of Sound, 2nd ed. §§ 357, &c.