PAPER BY PROF. HELMHOLTZ. 115 



When the form of the wave-line is given, the values of the two func- 

 tions, ?/', as is well known, are completely determined by the above 

 given boundary conditions (lb) to [Ig) and in that case the two integrals, 

 which multiplied by one-half of the density of the respective fluids, give 

 the living forces, namely : 



"WlKtK'i;]- <^' 



aud 



become absolute minima for such variations of the functions y-i, as are 

 possible under the given circumstances, when at the same time the 

 values pi and p2 are considered as invariable. 



Ou the other hand the form of the wave-line is not yet determined 

 b}^ the conditions hitherto given, except in so far that it must be period- 

 ical with the period A. We can however determine the form of this 

 boundary line corresponding to the physical condition that the pressure 

 shall be the same ou either side of it, in that we require that the varia- 

 tion of the difference between the potential energy ^ and the living 

 force L=Li+Lz sball disappear, or 



6[0-L]=O , . . . . {2b) 



The potential energy depends upon the unequal elevation of the dif- 

 ferent parts of the surface of heavier fluid above the level surface x=0. 

 Its amount is easily seen to be given by the equation 



^=y{S2-s,)/x'cly (2c) 



If Sz is the denser fluid, then the positive x, as already remarked, 

 must be assumed as ascending perpendicularly and y must be taken as 

 a positive quantity. 



When the linear element d s of the boundary-line of the two fluids is 

 displaced upwards normal to its own direction b^' the infinitely small 

 quantity S JV, then the variation becomes 



6 ^=(j{s2-Si)J'xSy. ds {2d) 



The variation of L can be executed in two steps. In the first of these 

 we imagine the boundary-line displaced in the above-given manner and 

 first allow the two functions if-i and y-g in each point of space to remain 

 unchanged, but in doing so, on that side where space is gained by the 

 displacement d s, imagine this strij) so gained to be filled with the con- 

 tinuous prolongation of the ip that pertains to this side, and so that the 

 equation J ^'=0 continues to be satisfied in that region [and so that the 

 prolongation of ?/? just mentioned enters here instead of the value of the 

 other function of if- previously existing here]. This prolongation of the 

 function ip into the strip just described is, as well known, only possible 

 in one manner without forming discontinuities. Only when a cusp of 



