1892.] Stability and Instability of Viscous Liquids. 275 



The remainder of the paper is occupied with the discussion of a 

 variety of problems relating to jets and wave motion. 



I find that when a cylindrical jet is moving through the atmosphere, 

 the tendency of the viscosity of the jet is always in the direction of 

 stability. The velocity of the jet does not affect the stability unless 

 the influence of the surrounding air is taken into account ; if, how- 

 ever, this is done, it will be found that it gives rise to a term propor- 

 tional to the product of the density of the air and the square of the 

 velocity of the jet, whose tendency is to render the motion unstable. 

 The tendency of surface tension (as has been previously shown by 

 Lord E-ayleigh) is in the direction of stability or instability accord- 

 ing as the wave-length of the disturbance is less or greater than the 

 circumference of the jet. 



If, in addition, the jet is supposed to be electrified, the condition of 

 stability contains a term proportional to the square of the charge 

 multiplied by a certain number, n. When the ratio of the circum- 

 ference of the jet to the wave-length is less than 0'6, n is positive, 

 and the electrical term tends to produce stability ; but when this 

 ratio is greater than 0*6, n is negative, and the electrical term tends 

 to produce instability. It mast, however, be recollected that when 

 the above ratio is greater than unity the tendency of surface tension 

 is to produce stability ; but if the influencing body is capable of in- 

 ducing a sufficiently large charge, the electrical term (when 2 TO, > X) 

 will neutralize the effect of surface tension and viscosity, and the 

 motion will be unstable. 



The well-known calming effect of " pouring oil on troubled waters " 

 has passed into a proverb. The mathematical investigation of this 

 phenomenon is as follows : The oil spreads over the water so as to 

 form a very thin film ; we may therefore suppose that the thickness 

 I of the oil is so small compared with the wave-length that powers of 

 I higher than the first may be neglected. Also, since the viscosity of 

 olive oil in C.G.S. units is about* 3 25, whilst that of water is about 

 0-014, the former may be treated as a highly viscous liquid, and the 

 latter as a frictionless one. 



The result is as follows : 



Let p lt p be the densities of the water and oil, TI the surface ten- 

 sion between oil and water, T the surface tension between oil and air, 

 fji the viscosity of the oil, and e kt the time factor, then, to a first 

 approximation, 



For olive oil, T! = 20'56, T = 36'9, so that T > T! ; and I find that 

 * Osborne Reynolds, ' Phil. Trans.,' 1886, p 171. 



