On the Stability of the Motion of a Viscous Fluid. 113 
with the same velocities along the boundary, then 
A=A + ^^(ny.dxdydz; .... (2) 
where A is the dissipation of energy in unit of time at the mode 
of motion M , A at the mode of motion M, O' the angular velo- 
city corresponding at any point of the region to the mode of 
motion indicated by M — M . 
Proof. " Let u, v' , w' represent the component velocities, 
A' the dissipation of energy by friction at the mode of motion 
M — M , then at every point of the region occupied by the 
fluid we have by definition, 
(3) 
V —V — v , 
w f = w—io . 
el Along the boundary, 
u f =0, v' = 0, w' = (4) 
" Substituting the values of u, v, iv from (3) in the well- 
known expression for the dissipation-function, 
2 
Cr/Sw Sv\ 2 , /hu Bwy , /Bv Su\ 2 i 7 7 7 
'^IKBy'Bz) + te~^J + {^-8y)] d * d y dz > 
we find 
A — A a. \f xA (T/^ w o ^ u> ^ u ° ^ u ' ^ u ° & u '\ 
J LV&c 8% By By Bz Bz ) 
/Bvq _ Bv' Bv Bv f Bvq Bv\ 
\8a Boa By By Bz Bz J 
. /Biv dw' , Bw Bw' Bw Bw'\~\ , 7 , 
+ \Ja7'J*7 + Jy---8y- + -8z---87)] d * d y dz 
— 9 ClY^ Bv \/Bu/ Bv'\ /Bu Biv \(Bu' Bu'\ 
^JLVfy ~Jz~)\8y~~ ~Bz) + \Bz~~B^AJz~~8^) 
