=vjc 
Bu „ Bv „„ 
i/ie Stability of the Motion of a Viscous Fluid. 117 
it takes the form 
(/o^/S 2 w B 2 v B 2 u B 2 u\ 
^ J U \Sx Bz + das By Bz 2 By 2 ) 
Bv/ B 2 u B 2 w _ B?v _ &v\ 
+ Bt\ByBz + ByBz Ba* Bz 2 ) 
Bio/ B 2 v B 2 u B\o B 2 w\ . , . 
+ WWBy- + Bz-^-Bf-W dxd V dz 
-k— X/ 2 w \ dx dy dz 
" Combining the values of both terms, we have 
" This expression remains negative, and therefore the dissipa- 
tion of energy is decreasing, till everywhere in the fluid 
Bu _ Bv _ Bio _ 
Bt~Bi~Ji ~ > 
but then the motion has become steady, and is necessarily 
identical with the motion represented by M ." 
Theorem TV.— The mode of motion represented by M is 
always stable as far as squares and products of velocities may be 
neglected. 
Proof. " Let the mode of motion M be disturbed by any 
cause whatever. Then the dissipation of energy by internal 
friction is necessarily increased (Theorem II.) ; but as soon 
as the cause of disturbance ceases it must decrease again 
(Theorem III.) till it reaches the value A , and then the mode 
of motion M is restored." 
From this theorem I draw the following conclusions: — 
1st. That the existence of unstable modes of fluid-motion 
and the origin of eddies cannot be explained without taking 
into account squares and products of velocities ; for that the 
equations (1) for steady motion with low velocities cannot 
lead directly to eddying fluid-motion, whatever the velocities 
along the boundary be, is a consequence of the well-known 
relations 
V 2 f=0, V 2 ^=0, V 2 £=0. 
According to these relations, f , rj, and £ can have no maxi- 
