120 THE MECHANICS OF THE EARTH's ATMOSPHERE. 



dimimitiou of tbe values pr and ^.V will necessarily make the larger 

 values of SLi and 6L2 with respect to (^^ disappear. Then variations 

 of d'P can not be counterbalanced by opposite and equal values of L 

 aud then can at the most ouly one limiting value exist, /. e., that which 

 corresponds to the plane surface. The limit for the smallest allowable 

 values of \\ and p2 results from the preceding investigation as follows :* 



Hence the range of values of {pifand {\y,Y that permit stationary waves 

 of the wave-length A is limited on its loiver side. 



It is to be noticed that the quantity pz determines the progressive 

 velocity of the wave with respect to the water; p], on the other hand, 

 determines the velocity of the wind relative to the wave. Either of 

 these can be small if the other is sufficiently large. 



II. THE THEOREM OF MINIMUM ENERGY APPLIED TO STATIONARY 

 BILLOWS WITH A CONSTANT VALUE OF THE VELOCITY POTENTIAL. 



The value of the living force, as given in equation (2), can by partial 

 integration be written 





in which the integral relates only to the upper horizontal boundary line. 

 The portions of the integral for the other limit of the space Sx all disap- 

 pear. Since now according to equation (1) 



dx dy 

 there results 



Of, if we put 



which difference is independent of .r, we obtain 



i^i=|Vli . (3) 



and similarly 



L,=.-J^^2 : . . (3rt) 



* My attention has been called to the fact that Sir William Thomson has already 

 given this eqnatiou as the first approximation, taking into consideration the streugih 

 of tlie wind, I'hilns. Ma;/., 1871 (4), vol. XL, p. 3G2, where, moreover, the influence of 

 capillarity is also considered. 



