a Sphere in a Viscous Fluid. 



331 



with the lower surface of a cover-glass could be measured. 

 The bubbles are so small that, although they are in contact 

 with a plane glass surface, they may be considered spherical. 

 The following results make up the first complete set ob- 

 tained, and are given for that reason only. The temperature 

 was about 15° C„ but was not specially noted. The velocities 

 were determined by timing an ascent through 66 centim. 



Table I. 

 Air-bubbles in Water. 



Radius 1 

 in centim. J 



Velocity 1 

 in cm. per sec. J 



•0071 

 0-92 



•0076 

 1-16 



•0085 

 113 



•0107 

 1-59 



•0109 

 1-70 



•0134 

 216 



Radius "1 

 in centim. J 



Velocity 1 

 in cm. per sec. J 



•0141 

 2-59 



•0180 

 3-14 



•0233 

 4-40 



•0256 

 4-80 



•0381 



7-89 



•0385 



7-47 



In the last three observations recorded in the table the 

 radius of the bubble was determined by collecting a large 

 number of equal bubbles and finding the volume they 

 occupied. 



The results are plotted in fig. 3 (p. 332) with the radius as 

 abscissa and the velocity as ordinate. In the same diagram 

 are shown the theoretical results for a solid sphere falling 

 through water at 15°, assuming <r=2p. The parabola for 

 which /3 = co corresponds to the case of no slip, and that for 

 which /3=0 to the case of infinite slip. The " critical" points 

 are marked by stars. 



It will be seen that the observed points lie on a straight 

 line cutting the line of zero velocity not far from the origin. 

 Hence 



Y = K{a-b) 

 where b is a small constant quantity. 



For velocities greater than the critical velocity the terminal 

 velocity of small bubbles is proportional to the radius 

 diminished by a small constant. 



If we employ the principle of dynamical similarity in 

 the manner already illustrated, to determine the form of the 

 quantity K, we obtain 



\ P J ** ' 



