530 Lord Rayleigh on Convection Currents in 



are vertical, and the circulation in each cell approximates to 

 that already indicated. This phase is brief (1 or 2 seconds) 

 for the less viscous liquids (alcohol, benzine, &c.) at ordinary 

 temperatures. Even for paraffin or spermacetti, melted at 

 100° C, 10 seconds suffice ; but in the case of very viscous 

 liquids (oils, &c), if the flux of heat is small, the deforma- 

 tions are extremely slow and the first phase may last several 

 minutes or more. 



The second phase has for its limit a permanent regime of 

 regular hexagons. During this period the cells become equal 

 and regular and allign themselves. It is extremely pro- 

 tracted, if the limit is regarded as the complete attainment 

 of regular hexagons. And, indeed, such perfection is barely 

 attainable even with the most careful arrangements. The 

 tendency, however, seems sufficiently established. 



The theoretical consideration of the problem here arising 

 is of interest for more than one reason. In general, when a 

 system falls away from unstable equilibrium it may do so in 

 several principal modes, in each of which the departure at 

 time t is proportional to the small displacement or velocity 

 supposed to be present initially, and to an exponential factor 

 e qt , where q is positive. If the initial disturbances are small 

 enough, that mode (or modes) of falling away will become 

 predominant for which q is a maximum. The simplest 

 example for which the number of degrees of freedom is 

 infinite is presented by a cylindrical rod of elastic material 

 under a longitudinal compression sufficient to overbalance 

 its stiffness. But perhaps the most interesting hitherto 

 treated is that of a cylinder of fluid disintegrating under 

 the operation of capillary force as in the beautiful experi- 

 ments of Savart and Plateau upon jets. In this case the 

 surface remains one of revolution about the original axis, 

 but it becomes varicose, and the question is to compare the 

 effects of different wave-lengths of varicosity, for upon this 

 depends the number of detached masses into which the 

 column is eventually resolved. It was proved by Plateau 

 that there is no instability if the wave-length be less than 

 the circumference of the column. For all wave-lengths 

 greater than this there is instability, and the corresponding 

 modes of disintegration may establish themselves if the 

 initial disturbances are suitable. But if the general dis-" 

 turbance is very small, those components only will have 

 opportunity to develop themselves for which the wave- 

 length lies near to that of maximum instnbility. 



It has been shown * that the wave-length of maximum 



* Proc. Lond. Math. Soc. vol. x. p. 4 (1879) ; Scientific Papers, vol. i. 

 p. 361. Also ' Theory of Sound,' 2nd ed. §§ 357, &c. 



