758 Mr. H. D. Arnold on Stokes's Law for 



with radii so great that Stokes's law no longer held even 

 approximately, they do not serve to define the region of 

 validity of the simple formula. 



As to the existence of surface slip, experimenters seem about 

 evenly divided. In general, however, the more sensitive the 

 experimental method used, the smaller the assigned value of 

 the coefficient has become. If, instead of the coefficient of 

 slip, we use the coefficient of sliding friction, the form 

 of Stokes's law becomes* 



i 9/* L ^'ftr + Zpl 



which for ft infinite reduces to the original form, but for ft 

 zero gives a value 50 per cent, higher. We see at once that 

 if ft has a finite value, even though it be very large, this 

 correction factor should become appreciable tor a sphere of 

 sufficiently small radius in a liquid of high viscosity. This 

 gives a very sensitive method of detecting the presence of 

 slip, somewhat similar to that used by Whethamf in the 

 analogous case of flow through tubes. It has hitherto 

 suffered under the disadvantage that no experimental veri- 

 fication of this theoretical correction factor has seemed 

 possible. The proof of the correctness of the factor in the 

 case of air bubbles, however, where, as will be shown later, 

 the apparent value of ft may be nearly zero, completely 

 removes this objection, and allows the use of the formula to 

 determine ft. 



Condition (5) is imposed to simplify the mathematical 

 analysis, since it allows us to neglect terms of the order of 

 the square of the velocity. These terms have been called 

 the inertia terms. It can be shown that the solution may be 

 expected to hold only when the velocity is small compared 



with -^J. The value of r for which v= -■ is called the 

 pr pr 



critical radius, and will be designated by r. Various ob- 

 servers have assumed that the upper limit of radius for which 



the simple formula may be used is of the order — (Laden- 

 berg), to -4= (Zeleny). 



It is the first object of this experiment to find the upper 

 radius limit of applicability of the simple law in terms of r. 



* Basset, Treatise on Hydrodynamics, vol. ii. 

 t Proc. Roy. Soc. xlviii. p. 225 (1890). 

 X Lamb, Hydrodynamics. 



