a Sphere in a Viscous Fluid* 333 



The value of the constant k must be *518 if the line drawn in 

 the diagram is to be represented by the formula. We may 

 simplify the result by assuming k = ^, a course only justified 

 by the fact that later observations agree sufficiently well with 

 the simplified formula. 



Again the constant b must be of the dimensions of a length. 

 Now the only length available for comparison is the critical 

 radius a. Hence we may suppose b expressed as a certain 

 fraction of the critical radius, whose value at 15° is '0085 

 centim. From the diagram we find b—^a, nearly. Thus 



v=i (my 



a — ia 



One difficulty in determining the velocity of a bubble by 

 this method has not yet been alluded to. This arises from the 

 fact that the volume of the bubble does not remain constant. 

 Two causes will be active in changing the volume of a bubble 

 of gas ascending through a liquid — the change in pressure and 

 the gradual solution of the gas in the liquid. In the case of small 

 bubbles the second cause generally predominates. Using 

 ordinary distilled water it was found that for velocities 

 greater than about 1 centim. per second there was practically 

 no difference between the times for the first and second 

 half of an ascent of 72 centim , but for velocities less than 

 this the second half of the journey took longer th;m the first. 

 Extremely small bubbles were dissolved before travelling the 

 whole distance. 



1 sought to overcome this difficulty by bubbling air through 

 the water used for a long period so as to get a saturated 

 solution. However, the most satisfactory series of results 

 was obtained on a day on winch the temperature fell to about 

 8° C. and so gave rise to sufficiently complete saturation. 



The results are given in Table II. 



The calculated values for the velocity are given in two 

 columns. In the first are the values calculated from the 

 formula of Stokes (Parabolic formula), in the second are those 

 found from the empirical relation given above (Linear 

 formula). The critical radius, assuming v — "01404 at 8°'4 C, 

 is equal to "00967 centim. 



On comparing the calculated and observed results it will 

 be seen that for bubbles less than the critical size the velocities 

 tend to agreement with the theoretical values ; for larger 

 bubbles the velocities agree with those given by the linear 

 formula. 



Pldl. Mag. S. 5. Vol. 50. No. 304. Sept. 1900. 2 A 



