1256 
outer boundary, as is shown in Fig. 2. Similar motions 
are produced in molten wax and in molten metals 
cooled from above, just as in any liquid cooled from 
above. Thus Bénard’s discovery of convection cells 
was made, during experiments with liquid coherers 
containing metallic particles, in early experiments on 
wireless reception. 
The clearly defined convection cells shown in Fig. 2 
do not form immediately. When the volatile liquid is 
poured into a dish the liquid at first appears to be 
violently disturbed, showing no recognisable pattern. 
If the liquid is exposed to a light current of air, or if, 
even in still air, it is left uncovered, the violently dis- 
turbed state continues for a long time. When the vessel 
containing the volatile liquid is covered by a sheet of 
glass placed above the surface, but not in contact with 
the edges of the vessel, the free evaporation of the 
liquid is checked, and the cells as shown in Fig. 2 are 
quickly formed and retain their general form for an 
almost indefinite period. 
Similarly, in the experiments on polygonal convec- 
tion cells in air, described below, there is an initial 
stage in which the air is divided into polygonal cells 
of four, five, or six sides. These cells are unsteady, and 
eventually settle down to a fairly steady pattern with 
five or six sides. 
The Theoretical Discussions of Lord Rayleigh and 
Others 
Lord Rayleigh [14] was led to examine the physical 
conditions leading to the formation of the convection 
cell by seeing Bénard’s description of his experiments, 
in some of which the cells were all truly hexagonal. 
Rayleigh started from the hypothesis that the veloci- 
ties, and the departures of pressure and temperature 
from their initial values, were all of the form e*% e*! 
e'77 ¢', and that the motion with the highest in- 
crement would rapidly predominate over all others. 
He found a new criterion, showing that no motion 
will occur in a “statically” unstable liquid, unless 
= 271 k: 
pr z Po us T a 
p 4gh* 
Where p:, po, and p are the densities at the top and 
bottom of the layer and the mean density within the 
fluid, respectively; x is the coefficient of thermometric 
conductivity; v the kmematic coefficient of viscosity; 
and h the depth of the layer. 
Thus for any given depth of layer there is a finite 
value of the difference of density at top and bottom 
of the layer which must be exceeded before motion 
can occur. Moreover, the shallower the layer, the 
greater is the difference of density required to produce 
motion, while for a given difference of density there is 
a lower limit to the depth in which motion can occur. 
This last feature is readily displayed im any suitable 
liquid. If, for example, a cheap gold paint is used as a 
medium, and the containing vessel is tilted so that at 
one side the depth decreases to zero, the diameter of 
the cells will be greatest at the deeper edge and will 
decrease with decreasing depth, but only to a finite size, 
LABORATORY INVESTIGATIONS 
beyond which there will be a shallow layer showing no 
cellular motion. 
Bénard [2] showed that Rayleigh’s theory gave, for a 
circular cell, the value of 3.285 for the ratio of diam- 
eter of cell to depth of fluid. Bénard’s own experiments 
on spermaceti gave values of 3.27 to 3.34 for this 
ratio. 
Rayleigh treated the case when the fluid is limited 
by two free surfaces. Two other cases require con- 
sideration, that of two rigid boundaries, and that of 
one free surface, the other surface being a rigid bound- 
ary. The constant A in the criterion 
pi — po _ Ay 
p gh’ 
has been evaluated for different cases by Jeffreys [9] 
and more recently by Pellew and Southwell [12]. The 
latter writers give the following values of A: 
A 
1. Two free surfaces 657.5 
2. Two rigid boundaries 1708 
3. One free, one rigid surface 1101 
Rayleigh gave for case (1) the same value of A, 
4 
equal to = or 657.5. Jeffreys [9] gave 1709.5 for 
case (2). The detailed methods of computation used 
by these authors differed somewhat, and so the values 
given above may be accepted as reasonably accurate. * 
The theoretical treatment by all the above-men- 
tioned authors is based on the assumption that the com- 
ponents of velocity are sufficiently small to justify the 
neglect of their squares or products. This is a limitation 
which may lead to some uncertainties in the comparison 
of theory and experiment. Rayleigh further pomts out 
that in his treatment of the problem it is tacitly as- 
sumed that (p; — po)/p is small. If, then, the depth of 
the unstable fluid is small, the critical value of (p; — 
po)/p as defined by ae becomes large, and the inter- 
pretation of the criterion as stated by Rayleigh is open 
to doubt. When the fluid is a gas and the instability is 
produced by heating the gas from below, it becomes 
impossible to satisfy the criterion for small depths of 
the unstable layer, and polygonal cells of convection 
will not then occur. 
We have therefore to embark upon the consideration 
of small-scale laboratory experiments bearing in mind 
that the theory of Rayleigh and of others is not likely 
to include all phenomena that may occur. 
Motion in Unstable Layers of Air Having No General 
Motion 
A convenient apparatus for studying the motions in 
unstable air can readily be constructed, having as base 
a smooth metal plate, fitting on top of a shallow box 
containing a series of parallel glass rods on which a 
thin wire is wound. The base plate is heated by the 
passage of an electric current through the wire. The 
top of the chamber consists of a plane sheet of 
glass which forms the base of a vessel that can be 
filled with water to a sufficient depth to act as a check 
