14 MY. F. T. Tronton. On the Motion under [June 1, 



The surface of the bubble moves by no means in a rigid manner, 

 as can be seen by watching the movements of little particles of dust 

 which may be present. The liquid of the bubble is seen in constant 

 circulation flowing up the side with the current of glycerine, and 

 returning down the centre of the bubble. Thus the viscosity of the 

 liquid of which the bubble is composed must affect the velocity of its 

 descent, but in what follows it has been neglected ; this could be done, 

 because the viscosity of the liquid of the column in most of the 

 experiments was very great compared with that of the bubble. 



Collecting the various things on which the velocity may depend, 

 we have : 1, the pressure gradient, that is to say, the difference in 

 density of the two liquids multiplied by the acceleration of gravity 

 $g ; 2, the viscosity of the liquids ft ; 3, the surface tension between 

 the liquids S ; 4, the diameter of the tube D. It is difficult to see 

 that anything else could come in to affect the rate of flow unless it be 

 a slipping over the solid surface. 



Thus we may put 



V = 



Assuming the function to have the form 



V- 1 = 2A (#tyvS w D")* 



we can obtain three equations from the considerations of dimensions 

 to help determine the unknown exponents. 



From length,f 1 = 3x + yz + n, 

 From mass, = x + z + m, 



From time, 1 = 2y z 2m. 



Now if we suppose y = x, as may very well be done, seeing that the 

 flow is of a purely viscous nature, we are left with but one unknown, 

 on account of peculiarities in the coefficients. 



3 = 1, m (oj + l), n = 2x. 



Thus V- 1 = 2 A 



Since the velocity increases with difference in density of the liquids, 

 we give x the successive values 1, 2, 3, &c., and obtain the 

 velocity expressed in a series. 



As there are two coefficients of viscosity to be taken into account, 

 the series should properly be of the form 



* The form of the series represents the reciprocal, instead of the velocity itself, 

 because it so happened the constants were originally so calculated, and a change 

 would involve the arithmetical labour over again. 



t The dimensions of S are S = M/L 3 , of g = L/T 2 , of /i = M/LT, and of S = M/T 2 . 



