Flow of Viscous Liquids. 355 



the steady motion (M ) of an incompressible fluid moving 

 with velocities given at the boundary, less energy is dissi- 

 pated than in the case of any other motion (M) consistent 

 with the same conditions. And if the motion M be in pro- 

 gress, the rate of dissipation will constantly decrease until it 

 reaches the minimum corresponding to M . It follows that 

 the motion M is always stable. 



It is not necessary for our purpose to repeat the investiga- 

 tion of Korteweg ; but it may be well to call attention to the 

 fact that problems in viscous motion in which the squares of 

 the velocities are neglected, fall under the general method of 

 Lagrange, at least when this is extended by the introduction 

 of a dissipation function*. In the present application there 

 is no potential energy to be considered, and everything de- 

 pends upon the expressions for the kinetic energy T and the 

 dissipation function F. The conditions to be satisfied may 

 be expressed by ascribing given constant values to some of 

 the generalized velocities ; but it is unnecessary to introduce 

 more than one into the argument, inasmuch as any others 

 may be eliminated beforehand by means of the given relations. 

 Suppose, then, that yjr r is given. The other coordinates yfa, 

 i/r 2 , . . . may be so chosen that no product of their velocities 

 enters into the expressions for T and F, although products 

 with -^rr, such as ^ yfr r , will enter. These coordinates are, 

 in fact, the normal coordinates of the system when yjrr is con- 

 strained to vanish. Thus simplified F becomes 



V=fofa + ... + ib.+.* + ... + b„h+r + ... . (1) 



and a similar expression applies to T with a written for b. 

 Lagrange's equation is now 



«. t, + Ci rs fr + K^s + K fr=0, 



i/r s being any one of the coordinates yfr ly -ty 2} ... In this 

 equation yjr r =0, and ty r has a prescribed value; so that 



Cts^s + hf^-Ksfr ..... (2) 



is the equation giving \jr s . The solution of (2) is well known, 

 and it appears that ^ s settles gradually down to the value 

 given by 



bs^r s =—br s ^rj (3) 



since a S) b 8 are intrinsically positive. Further, 



<7F 



-^T = t{b s ^jr s ^ s + b r3 (yjr s f r + yjr s ^) }, 



* Theory of Sound, §81. 



