532 Lord Rayleigh on Convection Currents in 



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 io, 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 dis- 

 turbances 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 Boussinesq, who has applied them to one or two 

 particular problems. The special limitation which charac- 

 terizes them is the neglect of variations of density, except in 

 so far as they modify the action of gravity. Of course, such 

 neglect can be justified only under certain conditions, which 

 Boussinesq has discussed. They are not so restrictive as to 

 exclude the approximate treatment of many problems of 

 interest. 



When the fluid is inviscid and the higher temperature is 

 below, all modes of disturbance are instable, even when we 

 include the conduction of heat during the disturbance. But 

 there is one class of disturbances for which the instability is 

 a maximum. 



When viscosity is included as well as conduction, the 

 problem is more complicated, and we have to consider 

 boundary conditions. Those have been chosen which are 

 simplest from the mathematical point of view, and they 

 deviate from those obtaining in Benard's experiments, 

 where, indeed, the conditions are different at the two 

 boundaries. It appears, a little unexpectedly, that the equi- 

 librium may be thoroughly stable (with higher temperature 

 below), if the coefficients of conductivity and viscosity are 

 not too small. As the temperature gradient increases, in- 

 stability enters, and at first only for a particular kind of 

 disturbance. 



The second phase of Benard, where a tendency reveals 

 itself for a slow transformation into regular hexagons, is not 



