356 Lord Rayleigh on the 



in which the summation extends to all values of s other than 

 r. In this i|r,. = 0, so that 



5=S+.{ft.+. + fc.+r}«-Sa.+. f , . . (4) 



by (2) . The last expression is intrinsically negative, proving 

 that until the steady motion is reached F continually de- 

 creases. Korteweg's theorem is thus shown to be of general 

 application to systems devoid of potential energy for which 

 T and F can be expressed as quadratic functions of the 

 velocities with constant coefficients. 



It may be mentioned in passing that a similar theorem 

 holds for systems devoid of kinetic energy, for which, however, 

 F and V (the potential energy) are sensible, and may be 

 proved in the same way. If such a system be subjected to 

 given displacements, it settles down into the configuration of 

 minimum V ; and during the progress of the motion V 

 continually decreases. 



The theorem of Korteweg places in a clear light the general 

 question of the slow motion of a viscous liquid under given 

 boundary conditions, and the only remaining difficulty lies in 

 finding the analytical expressions suitable for special problems. 

 It is proposed to consider a few simple cases relating to 

 motion in two dimensions. 



Under the above restriction, as is well known, the motion 

 may be expressed by means of Earnshaw's current function 

 (yjr), which satisfies 



V 4 *=0, (5) 



the same equation as governs the transverse displacement of 

 an elastic plate, when in equilibrium. Of this analogy we 

 shall avail ourselves in the sequel. At a fixed wall yjr retains 

 a constant value, and, further, in consequence of the friction 

 dyjr/dn, representing the tangential velocity, is evanescent. 

 The boundary conditions for a fixed wall in the fluid problem 

 are therefore analogous to those of a clamped edge in the 

 statical problem. 



The motion within a simply connected area is determined 

 by (5) and by the values of the component velocities over 

 the boundary. If we suppose that two such motions are 

 possible, their difference constitutes a motion also satisfying 

 (5), and making i/r and difr/dn zero over the boundary. Con- 

 siderations respecting energy in this or in the analogous 

 problem of the elastic plate are then sufficient to show that 

 yjr must vanish throughout ; and an analytical proof may 



