IV. VISCOSITY MEASUREMENTS 113 



the sphere is falling under the influence of gravity, the force acting 

 on it is its apparent weight, that is to say, its volume times the dif- 

 ference between the specific gravity of the sphere, a, and the fluid, p, 

 times the gravity constant, (j. Thus: 



W = '^iwa^ia — p)g = Girr]nv 

 so that: 



T] = 2g(a — p)a^/9v 

 and: 



V = 2g((T - p)aV9T7 



Thus, if we have a sphere falling through a fluid and we know its 

 specific gravity, its radius, its speed, and the specific gravity of the 

 fluid through which the sphere falls, we can know the viscosity. 



In the original derivation of the law that bears his name, Stokes 

 assumed that certain mathematical terms, the so-called semi(iuad- 

 ratic terms, could be neglected. Actually, they can properly be neg- 

 lected only if the movement of the sphere is slow and its size small. 

 Thus Raj'leigh in 1893 pointed out that the assumption is warranted 

 only if vap/ri is negligible compared to unity. This condition is read- 

 ily met in studies in protoplasmic viscosity, for in protoplasm the 

 spheres that move are tiny, of the order of magnitude of 10~^ cm., 

 and their rate of movement very slow, typically 10~^ cm. per second. 

 Modern authors agree that the original form of Stokes' law holds if the 

 Reynolds number (2vap/r]) is small. Barr (3) states that, so long as 

 the Reynolds number is negligibly small compared with unity, the 

 original Stokes law holds; if the Reynolds number is less than 0.05, 

 Stokes' law is accurate to about 1%. Schiller (13) also states that 

 Stokes' law holds if the Reynolds' number is much less than one. At 

 higher speeds, when the Reynolds' number is larger, a more exact 

 formulation has been derived by Oseen. In the studies of protoplasm, 

 Oseen's law is of no advantage, and the original form of Stokes' law 

 can properly be used. 



The theoretical derivation of Stokes' law involves a number of 

 assumptions. These are that the motion of the sphere is slow and 

 that the motion is steady and free from acceleration, that there is no 

 slip between the sphere and the fluid, that the sphere is rigid, and 

 that the fluid is homogeneous and extends infinitely in all directions. 

 As Arnold (14) showed, these assumptions do not affect the validity 

 of the law for a small particle dropping through a viscous fluid (c/. 

 also the discussion given by Barr, 3). For narrow tubes, a correction 



