THE DYNAMICAL THEORY OF GASES. 69 



and by transformation of co-ordinates we obtain 



idv dw 



dw . du\ (127) _ 



fdu dv^ 

 \dy dxj 



These are the values of the normal and tangential stresses in a simple gas 

 when the variation of motion is not very rapid, and when p, the coefficient 

 of viscosity, is so small that its square may be neglected. 



Equations of Motion corrected for Viscosity. 



Substituting these values in the equation of motion (76), we find 

 du dp (d*u d*u d*ii\ ^ d (du dv dw\ 



with two other equations which may be written down with symmetry. The 

 form of these equations is identical with that of those deduced by Poisson"" 

 from the theory of elasticity, by supposing the strain to be continually relaxed 

 at a rate proportional to its amount. The ratio of the third and fourth terms 

 agrees with that given by Professor Stokesf. 



If we suppose the inequality of pressure which we have denoted by q to 

 exist in the medium at any instant, and not to be maintained by the motion 

 of the medium, we find, from equation (123), 



q^Ce-*** ..................................... (129) 



= <7<f? if r= -=!- = .................. (130); 



p 



the stress q is therefore relaxed at a rate proportional to itself, so that 



| = | ................................. (131). 



We may call T the modulus of the time of relaxation. 



* Journal de I'Ecole Polylechnigue, 1829, Tom. xm. Cah. xx. p. 139. 



t " On the Friction of Fluids in Motion and the Equilibrium and Motion of Elastic Solids," Cambridge 

 Phil. Trans. Vol. VHI. (1845), p. 297, equation (2). 



