84 



Lord Rayleigh on 



[May 15, 



I propose to pass in review the leading features of Plateau's theory, 

 imparting, where I am able, additional precision. 



Let us conceive, then, an infinitely long circular cylinder of liquid, 

 at rest,* and inquire under what circumstances it is stable, or unstable, 

 for small displacements, symmetrical about the axis of figure. 



Whatever the deformation of the originally straight boundary of 

 the axial section may be, it can be resolved by Fourier's theorem into 

 deformations of the harmonic type. These component deformations 

 are in general infinite in number, of every wave-length, and of 

 arbitrary phase ; but in the first stages of the motion, with which 

 alone we are at present concerned, each produces its effect indepen- 

 dently of every other, and may be considered by itself. Suppose, 

 therefore, that the equation of the boundary is 



r=a + « cosfe, (6), 



where a is a small quantity, the axis of z being that of symmetry. 

 The waveJength of the disturbance may be called X, and is connected 

 with h by the equation &=2t7-\, _1 . The capillary tension endeavours 

 to contract the surface of the fluid ; so that the stability, or instability, 

 of the cylindrical form of equilibrium depends upon whether the 

 surface (enclosing a given volume) be greater or less respectively 

 after the displacement than before. It has been proved by Plateau 

 (see also Appendix I) that the surface is greater than before dis- 

 placement if ka>l, that is, if \<2ira ; but less, if Jea<l, or \>2:ra. 

 Accordingly, the equilibrium is stable, if X be less than the circum- 

 ference ; but unstable, if \ be greater than the circumference of the 

 cylinder. Disturbances of the former kind, like those considered in 

 the earlier part of this paper, 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 displace- 

 ments and velocities) increases rapidly, being equally multiplied in 

 equal times. The coefficient (q) of the time in the exponential term 

 (e^) may be considered to measure the degree of dynamical in- 

 stability ; its reciprocal q ~ 1 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 deter- 

 mined 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- 



* A motion common to every part of the fluid is necessarily without influence 

 upon the stability, and may therefore be left out of account for convenience of con- 

 ception and expression. 



