Mr Milne, Tensor Form of the Equations of Viscous Motion 345 



of the third rank whose components are zero when any two 

 of the suffixes are equal, and equal to ± ^yg when all three are 

 unequal, the sign being given by the parity of the number of in- 

 versions in V, p, a. If ^1, X2, a-3 are the co-ordinates, and if the 

 element of surface is denoted in the more usual way by dSi, dS2, 

 dSs, then the iCj-component of (1) reduces to 



VgiP^^dS^ + P'^dS^ + P'^^dSs). 



The "mean pressure" p is the invariant ^g^^ P'"". 



If k is the coefficient of viscosity, the usual equations* for the 

 pressures 



<,, fdu dv div\ -., dii\ 



_ _ , fdw dv\ 



'''" = ''- = * [di + dz) I 



^ (2) 



generahse into 



P"" = r" [- V - Ik {u%] + h [^r- (m^), + g^- {u^)^-\ ...(3) 



the notation (w^)<r denoting the covariant derivative; and the 

 equations of motion in terms of the pressures"}" 



^ Bt ^ ^ dx ^ dy ^ dz ^*^ 



generahse into p (-^~ - xA = {P'''')^ (5) 



where the generalisation of "differentiation following the motion" 

 is given by 



1)7 ^ ar + '"'"">'' • "" 



Substituting for P^^" in (5) and remembering that the covariant 

 derivatives of the ^'s are identically zero we find 



(7) 



where in the last term v and a have been interchanged as being 

 dummies. These are the differential equations satisfied by the 

 velocities. 



On the other hand the usual equations for the velocities ob- 

 tained by combining (2) and (4) directly, namely, 



* Lamb, Hydrodynamics, fourth edition, p. 570. t Loc. cit. p. 572. 



