the Stability of the Motion of a Viscous Fluid. 115 
the first may be reduced to 
- 2 ix J (V V V + v' V V + u/VV) dx dy dz, 
the second to 
+ fi j" (m'VV + 1/ V V + «/ V V) da dy ds; 
but then it is obvious that A' may be put at choice under 
one of two forms: — 
A'= —fi j(V V V + tfW + «/VV) da? dy d«, . (8) 
or 
+ (i " |-) 2 ] ^^=M(ao 3 ^^^.'' (9) 
Theokem II. — TAe mode of motion M is swc/t £/ia£ ^7te dis- 
sipation of energy by internal friction is the least possible con- 
sistent roith the same velocities along the boundary. It is unique 
in every simply connected region when these velocities are given. 
Proof. " The first part of this theorem follows immediately 
from the equation (2), the second term of the right-hand side 
of this equation being necessarily positive or zero. 
" Now let M' be a second mode of motion, consistent with 
the given velocities along the boundary and satisfying the 
equations (1). Then, as in both motions the dissipation of 
energy must be the least possible, A' is equal to A . This 
can be so only when, at every point, 
O'=0; 
but then M' — M should represent an irrotational motion 
with zero -velocities all over a closed boundary; and such a 
motion is known to be impossible." 
Theokem III. — When in a given region occupied by viscous 
incompressible fluid, there exists at a certain moment a mode of 
motion M, which does not satisfy the equations (1), then, the 
velocities along the boundary being maintained constant, the 
change which must occur in the mode of motion will be such 
(neglecting squares and products of velocities) that the dissipa- 
tion of energy by external friction is constantly decreasing till it 
reaches the value A and the mode of motion becomes identical 
with M . 
Proof. " The change occurring at any moment in the mode 
of motion is determined by the equations 
