362 MATHEMATICAL APPENDICES 6* 



These formulae determine the pressure exerted by the half of 

 the medium which is nearer the negative coordinates, and has 

 hitherto been called the first half, on the second, which lies on 

 the positive side. The action of the second half on the first is 

 expressed by the same formulae with changed sign. 



The first three of these six formulae give the pressures which act 

 normally on the stressed surface, i.e. the normal pressures, while 

 the last three express the magnitude of the tangential pressures 

 whose directions lie along the surface itself. 



7*. Interpretation of the Formulae 



Both forms of pressure are expressed by integrals which, from 

 the meaning of the function F, are easily seen to represent probable 

 mean values. 



Noting further that 



that is, the mass of gas contained in unit of volume, or its density, 

 we may write the foregoing formulae thus : 



Y y = p v* Z x = X z = p wu 



where the bar denotes the mean value of the magnitude placed 

 under it. 1 



These formulae, which were first given in such generality by 

 Maxwell 2 for the pressure in a gas in any state of motion that 

 does not depend on time or position, can be much, simplified 

 when there is only the heat-motion of the molecules and not a 

 forward motion of the gas as a whole. Since in this case the 

 motion is symmetrical in all directions of space, all functions 

 depending on uneven powers of the velocity-components u, v, w 

 vanish, and therefore 



vw wu = uv = 0. 



These terms also vanish when the direction of motion is along one 

 of the coordinate axes. 



1 [The author uses the notation M(x) to denote the mean value of x, but 

 the ordinary English custom is here followed. TB.] 



2 Phil. Mag. [4] xxxv. 1868, p. 195. 



