284] EXPRESSIONS FOR THE STRESSES. 513 



quantity is the ratio fi/p ; it is therefore usual to denote this by a 

 special symbol v, called by Maxwell the ' kinematic coefficient ' of 

 viscosity. The dimensions of v are [L 2 T~ 1 ]. 



The hypothesis made above that the stresses p xx , p xy ,. . . are linear functions 

 of the rates of strain a, 6, c,... is of a purely tentative character, and although 

 there is considerable a priori probability that it will represent the facts 

 accurately for the case of infinitely small motions, we have so far no assurance 

 that it will hold generally. It has however been pointed out by Prof. Osborne 

 Reynolds' 3 *' that the equations based on this hypothesis have been put to a 

 very severe test in the experiments of Poiseuille and others, to be referred to 

 presently (Art. 289). Considering the very wide range of values of the rates of 

 distortion over which these experiments extend, we can hardly hesitate to 

 accept the equations in question as a complete statement of the laws of 

 viscosity. In the case of gases we have additional grounds for this assump- 

 tion in the investigations of the kinetic theory by Maxwell f. 



The practical determination of p (or v} is a matter of some difficulty. 

 Without entering into the details of experimental methods, we quote a few of 

 the best-established results. The calculations of von Helmholtz|, based on 

 Poiseuille's observations, give for water 



0178 



in c.G.s. units, where 6 is the temperature Centigrade. The viscosity, as in 

 the case of all liquids as yet investigated, diminishes rapidly as the temperature 



rises ; thus at 17 C. the value is 



^ = 0109. 

 For mercury Koch found 



^ = 01697, and ^ 10 =-01633, 

 respectively. 



In gases, the value of /* is found to be sensibly independent of the pressure, 

 within very wide limits, but to increase somewhat with rise of temperature. 

 Maxwell found as the result of his experiments ||, 



H= -0001878(1 + -00366 <9) ; 



this makes /u proportional to the absolute temperature as measured by the 

 air-therrnometer. Subsequent observers have found a somewhat smaller 

 value for the first factor, and a less rapid increase with temperature. 

 We may take perhaps as a fairly established value 



= '000170 



* " 



On the Theory of Lubrication, &c.," Phil. Trans., 1886, Pt. I., p. 165. 



t "On the Dynamical Theory of Gases," Phil. Trans., 1867; Scientific Papers, 

 t. ii., p. 26. 



+ "Ueber Reibung tropfbarer Fliissigkeiten," Wien. Sitznngsber., t. xl. (1860); 

 Ges. Abh., t. i., p. 172. 



Wied. Ann., t. xiv. (1881). 



|| " On the Viscosity or Internal Friction of Air and other Gases," Phil. Trans., 

 1866; Scientific Papers, t. ii., p. 1. 



L. 33 



