PAPER BY PROF. HELMHOLTZ. 117 



. Wheu, therefore, the equation {2b) is to be satisfied, that is to say, 

 when we must have 



d { ^-i i =0 



then must p/=Pi throughout the boundary surface, which is the con- 

 dition of a stationary surface. 



The stability of the steady motion. — For any form of surface that nearly 

 corresponds to a stationary form, and which therefore still shows dif- 

 ferences of pressure, it lollows from the preceding- that such a surface 

 when it changes with the differences of the pressures experiences there- 

 fore a positive displacement rfiV where pi >i>i, therefore the quantity 

 {0—L) diminishes and consequently ai)proximates to a neighboring 

 minimum of ((?— X), and must therefore depart from the neighboring 

 maximum of the same quantity. 



The hydro-dynamic equations show in fact that the equality of pres- 

 sure in such cases can only be brought about by accelerations which 

 act in the direction from the stronger to the feebler pressure and 

 must disturb the steady motion. 



Therefore the stable equilibrium of a stationary waveform must 

 (among all possible variations of such a form) correspond to a minimum 

 of the quantity {0—L), just as in the polycyclic systems for a constant 

 velocity of their cyclic motions. When on the other hand this same 

 quantity (^ — L) attains a maximum value or a cusp value for some 

 other form of curve, then the condition of equality of pressure on both 

 sides of the boundary" surface is at least temporarily fulfilled; but 

 individual or the very smallest disturbances of the form of equilibrium 

 must continue to increase : the equilibrium will thus become unstable 

 as is actually recognized in natural water-waves by the foaming and 

 breaking of the crests of the waves. 



On the other hand it is to be remarked that these propositions hold 

 good only when the functions Li and Li are determined as minima in 

 accordance with the boundary conditions of the spaces within which 

 they hold good, and for every variation in the form of the boundary 

 line the functi(ms experience a change in accordance with this condi- 

 tion that they shall be minima. 



rjnder the assumptions already made, the function ^ is certainly 

 positive and fiuite, since only a finite quantity of liquid is present 

 which can be raised up only through the finite altitude Ex. L is also 

 necessarily positive but can become +go, since the summit of the wave 

 can approximate to the upper but the trough of the wave to the lower 

 boundary surface and the total constant quantity of moving fluid must 

 then be pressed with infinite velocity through infinitely narrow crevices. 



The quantity {<P—L) must therefore have a positive value for plane 

 boundary surfaces where ^=0, and it can become — cc for increasing 

 wave altitudes. Whether a minimum occurs between these limits, and 

 for what value of p this could occur, can only be decided by investigation 



