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XVII. On a General Theorem of the Stability of the Motion 
of a Viscous Fluid. By D. J. Korteweg, Professor of 
Mathematics at the University of Amsterdam* . 
THE interesting experimental investigation of Prof. 0. 
Reynolds! on sinuous fluid-motion and the origin of 
eddies induces me to communicate a very general theorem on 
the stability of fluid-motion, which I published some months 
ago in the Transactions of the Royal Academy of Amsterdam. 
Conceiving that the origin of eddies was to be explained by 
the existence of unstable solutions of the equations of motion, 
I endeavoured to find such solutions by means of the well- 
known equations for slow motion of viscous fluid, till I found 
out that in any simply connected region, tvhen the velocities along 
the boundary are given, there exists, as far as the squares and 
products of the velocities may be neglected, only one solution of 
the equations for the steady motion of an incompressible viscous 
fluid, and that this solution is ahvays stable. 
The first part of this theorem is due to Helmholtz}. He 
shows it to be a very simple consequence of another theorem, 
stating that " if the motion be steady, the currents in a viscous 
fluid are so distributed that the loss of energy due to viscosity 
is a minimum, on the supposition that the velocities along the 
boundaries of the fluid are given.'''' 
Though my demonstration of this theorem is somewhat less 
general than that of Helmholtz, I will write it down here, as it 
leads to a very simple and symmetrical expression for the dif- 
ference between the real dissipation of energy by internal 
friction with any motion and the minimum dissipation con- 
sistent with the same velocities at the boundary, which expres- 
sion may be useful in other cases. 
Theorem I. — Let M represent a mode of motion of an in- 
compressible fluid answering to the equations 
(i\7 2 Uo = 
S(Vp+/>o) ; 1 
8x 
, vX= S(V^>o), 
Ci) 
Sz ' J 
M another mode of motion of incompressible fluid, consistent 
* Communicated by the Author. 
t Proceedings of the Royal Society, vol. xxxv. 1883, p. 84. 
t Verh. des naturh.-med. Vermis zu Heidelberg, Bd. v. S. 1-7; Collected 
Works, i. p. 223. 
