296-297] GENERAL THEOREMS. 537 



by Art. 286. If the extraneous forces X, F, Z have a single-valued potential, 

 this vanishes, in virtue of the equation of continuity, by Art. 42 (4). 



Under these conditions the dissipation in the altered motion is equal to 

 \\\$&amp;gt;dxdydz + \\\& dxdydz .......................... (vii), 



or 2 (F+F }. That is, it exceeds the dissipation in the steady motion by the 

 essentially positive quantity 2F which represents the dissipation in the 

 motion u , v , w . 



In other words, provided the terms of the second order in the velocities 

 may be neglected, the steady motion of a liquid under constant forces having 

 a single-valued potential is characterized by the property that the dissipation 

 in any region is less than in any other motion consistent with the same 

 values of u, v, w at the boundary. 



It follows that, with prescribed velocities over the boundary, there is only 

 one type of steady motion in the region*. 



2. If u, v, w refer to any motion whatever in the given region, we have 

 2/= JJJ* dxdydz 



= 2Stt(px X a+Py V i&amp;gt;+P*zC + 2p V3 f+2pz X sr + 2p xy h)dxdydz ...... (viii), 



since the formula (ii) holds when dots take the place of accents. 



The treatment of this integral is the same as before. If we suppose that 

 u, v, w vanish over the bounding surface 2, we find 



= -pJJJ(w 2 + 2 + M&amp;gt; 2 ) dxdydz + pSH(Xu + Yv+Ziv) dxdydz ...(ix). 

 The latter integral vanishes when the extraneous forces have a single- 

 valued potential, so that 



F= - P $jj(u 2 + v 2 + w 2 )dxdydz ..................... (x). 



This is essentially negative, so that F continually diminishes, the process 

 ceasing only when u=Q, v=0, 10 = 0, that is, when the motion has become 



Hence when the velocities over the boundary 2 are maintained constant, 

 the motion in the interior will tend to become steady. The type of steady 

 motion ultimately attained is therefore stable, as well as unique f. 



It has been shewn by Lord RayleighJ that the above theorem can be 

 extended so as to apply to any dynamical system devoid of potential energy, 



* Helmholtz, &quot; Zur Theorie der stationaren Strome in reibenden Fliissig- 

 keiten,&quot; Verh. d. naturhist.-med. Vereins, Oct. 30, 1868 ; JHss. Abh., t. i., p. 223. 



t Korteweg, &quot;On a General Theorem of the Stability of the Motion of a Viscous 

 Fluid,&quot; Phil. May., Aug. 1883. 



+ I.e. ante p. 526. 



