346 Mr Milne, Tensor Form of tlie Equations of Viscous Motion 



generalise into 



/Dw^ \ d 



p[-^-Xj = - g^" g- ijj - p {u%) + ir^ {u-U...{^) 



which differ from (7) by the absence of the term 



kg>^^[{u%.~{u%,-\. (10) 



Now the Riemann-Christofiel tensor B\^,„ is conveniently de- 

 fined by the identity 



where A^ is any covariant vector ; but it is easy to prove also that 

 if Af^ is any contravariant vector^ then \ 



{A^U~-{A^U = -A^B-,,,. . \ 



Contracting this by putting a = fx and summing, we have 



{A'^),^-{A>^),,= - Ai^Gp,. 



It follows that the term in question, (10), is simply 



- kg''" Gp^ UP, 



which vanishes, the space being Gahlean. Were one attempting, 

 however, to discuss viscous motion in non-Gahlean space, with 

 the generaUsations of (2) and (4) as a dynamical basis, one would 

 be led to an incorrect result by hastily generalising (8), although 

 this is merely a combination of (2) and (4); and the interest lies in 

 the circumstance that it is precisely the contracted Riemann- 

 Christofiel tensor that appears as an error term. 



