PAPER BY PROF. HELMHOLTZ. 113 



stable equilibrium of masses at rest, but with moviug masses that are 

 in steady motion. 



Some examples of such differences have indeed been already treated, 

 as in the rotation of a solid body about the axis of its greatest or least 

 moment of inertia, and in the rotation of a fluid ellipsoid subject to 

 gravity. But a general principle such as is giveu for bodies at rest, in 

 the proposition that stable equilibrium requires a minimum of poten- 

 tial energy, has never yet been established for a moving system of bodies. 



The following investigations lead to such propositions, which more- 

 over can also be considered as generalizations of the propositions that 

 1 have deduced from the general equations of motion given by Lagrange 

 in their application to the motion of " poly cyclic" systems.* 



I. THE THEOREM OF MINIMUM ENERGY APPLIED TO STATIONARY 

 WAVES HAVING A CONSTANT QUANTITY OF FLOW. 



As in my paper of last year,t I indicate by u and v the component 

 velocities of the particles of water during any motion that is free from 

 vortices by the equations : 



I again assume, whenever the opposite is not expressly stated, that 

 the coordinate system for x y is at rest with reference to the wave, x 

 being vertical, positive upward, y horizontal. Therefore the wave sur- 

 face is at rest with reference to these coordinates M'hile the two fluids 

 :flow steadily along it. The wave curve will be considered as i)eriodical 

 with the wave length A. On the other hand, the flowing fluid will be 

 considered as bounded by two horizontal planes whose equations are 



x=Hi and x=—E2 (la) 



Corresponding to this, I indicate the remaining quantities that refer 

 to the fluid which is on the positive side of x by the subscript 1 ; those 

 that are on the negative side of x by the subscript 2. 



The wave-lines and these two horizontal boundary lines must be 

 stream lines — that is to say, ?/' must have a constant value throughout 

 their whole length. Since each of the functions ^' can contain an arbi- 

 trary additive constant, therefore we can assume arbitrarily both of 

 the values of ip for one of the stream lines. I assume that for the wave 

 fine for which 



X = X 



we have the value 



^ = (lb) 



* Eronecker tmd TTtyo'strass, Journ.fiir Mathemai., iy.^4, vol", xcvii, p. 118. 

 t [See the previous paper, No. VI, iu this collection of Translations. ] 



80 a 8 



