274 Mr. A. B. Basset. [Nov. 17, 



ties. If e w be the time factor of a disturbance of wave-length X, the 

 value of k is 



_ +w ............ 



\ par \ V ) 



where n is a root of the equation Ji(%) = 0. 



Experiment shows that when the velocity is greater than about 

 6 inches per second, the frictional tangential stress of water in con- 

 tact with a fixed or moving solid is approximately proportional to the 

 square of the relative velocity. This introduces a constant ft, which 

 may be called the coefficient of sliding friction, whose dimensions are 

 [MLr 3 ], and are therefore the same as those of a density. This 

 constant may have any positive real value ; ft = corresponding to 

 perfect slipping or zero tangential stress, whilst ft = oo corresponds 

 to no slipping, which requires that the velocity of the liquid should 

 be the same as that of the surface with which it is in contact. Owing 

 to the intractable nature of the general equations of motion of a 

 viscous liquid, I have been unable to obtain a complete solution, 

 except on the hypothesis that ft is an exceedingly small quantity. 

 This supposition, I fear, does not represent very accurately the actual 

 state of fluids in contact with solid bodies ; but, at the same time, the 

 solution clearly shows that the instability observed by Professor 

 Reynolds does not depend upon viscosity alone, but is due to the 

 action of the boundary upon a viscous liquid. 



To a first approximation, the real part of Jc is proportional to 



Waft (w* + mV) a /Vk 



-is*- ................. (2) ' 



where 2?r/m is the wave-length of the disturbance, and n is a root of 

 the equation Ji(W) = 0. Since the second term is a number, this 

 shows that the motion will be stable, provided 



Waft/fi < a number. 



The experiments of Professor Reynolds conclusively show that the 

 critical velocity at which instability commences is proportional to 

 ^</a; and the fact that the theoretical condition of stability turns out 

 1o be that Wo//t, multiplied by a quantity of the same dimensions as 

 a density, should be less than a certain number, appears to be in 

 substantial agreement with his experimental results. 



The results of the investigation may be summed up as follows : 



(i.) The tendency to instability increases as the velocity of the liquid, 

 the radius of the tube, and the coefficient of sliding friction increase ; 

 out diminishes as the viscosity increases. 



(ii.) The tendency to instability increases as the wave-length (2ar/m) 

 of the disturbance increases. 



