82 Prof. A. H. Gibson on the Stability of 



For such a fluid, subject to no external forces, these 

 equations may be written 



9 Tt = ~ { dx^ + pVu) + dy (PM + P vv )+dz( p ** + pVW> > } V (1) 



and 



dw , dv dltf 

 a^? ay dz 



(2) 



Multiplying respectively by u, v, w, integrating, adding, 

 and writing 



the rate of increase of kinetic energy per unit volume is 

 given by 



( ( L + U A + v — + A\j$ = 

 \dt dx dy dz) 



Tx (up xx ) +^( mx ) + j z (up zx ) 



- < + I («P*) + Jy ^ + dz tyJ > 



da; 

 du 



Pxx Tx 



dv 



du 

 dy 

 dv 



+P* T +P~ Tz 



" r ^da ! +Pyy dy JrP2y dz 



dw 



dw 



I 



+ p**jz + P' x 'j;, + P" 



dy 



du 

 Tz 

 dv 



Tz 



dw 



dz ) 



y. . (3) 



The first term on the right of this equation represents the 

 rate at which work is being done by the surrounding fluid 

 per unit of volume, so that, for conservation of energy, as 

 pointed out by Stokes * the second term on the right must 



* Cambridge Phil. Trans, vol. ix. p. 57. 



