114 L. V. H E I L B R U N N 



is necessary, but such correction becomes unimportant if the diameter 

 of the tube is more than ten times that of the sphere (4) ■ 



The apphcation of Stokes' law to protoplasm is discussed by Heil- 

 brunn (1,15). In the study of protoplasmic viscosity, the spherical 

 particles that move through the protoplasm are sufficiently small and 

 their movement is slow. Nor is one ordinarily troubled by the need 

 for corrections due to the walls of the cell. On the other hand, there 

 is one serious difficulty. This is the fact that Stokes' law governs the 

 movement of a single sphere. The presence of more than one sphere 

 introduces complications. In protoplasm, there are numerous par- 

 ticles that move under the influence of gravity or centrifugal force, 

 and the velocity of any single particle tends to be slowed by the pres- 

 ence of other particles in its neighborhood. Various attempts have 

 been made to apply Stokes' law to the movement of large numbers 

 of spheres, and corrections to the law have been proposed by Cunning- 

 ham, Smoluchowski, and others. Of these corrections, the only one 

 that has been used in the study of protoplasm is that of Cunningham 

 (16). When the Cunningham factor is introduced into Stokes' law, 

 this becomes: 



2g(<T - pY 



V = 



in which: 



46(6* - a^) 



(6 - a)2(464 - h^a - Gb^a^ - ha^ - 4a4) 



a is the radius of the sphere; 6 is half the distance between t^yo ad- 

 jacent spheres. But, in making the computation, Cunningham f inds 

 it is better to substitute for 6 a quantity 6', which is equal to 6 V 3/2. 

 As a matter of fact, in most determinations of protoplasmic viscosity, 

 the biologist or biophysicist is interested in the changes of viscosity 

 under different conditions rather than in the absolute viscosity. For 

 measurements of relative viscosity, no attention need be paid to the 

 Cunningham correction. 



Gravity Method. Because of the fact that in cells generally the 

 particles do not fall under the influence of gravity, the gravity 

 method has been used only rarely. The technique of the method 

 is extremely simple. All that is necessary is to watch the fall of the 

 moving particles with a microscope. The microscope is set up so 



