I 



120 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



diminution of the values pi 2 and p 2 2 will necessarily make the larger 

 values of 6L y and dL 2 with respect to 6$ disappear. Then variations! 

 of 6<P can not be counterbalanced by opposite and equal values of /. 

 and then can at the most only one limiting value exist, i. e., that whicl 

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

 values of pi and p 2 results from the preceding investigation as follows:* 



Hereby the range of values of (^{f and (ty 2 ) 2 that permit stationary waves 

 of the icavelength A is limited on its lower side. 



It is to be noticed that the quantity p 2 determines the progressive 

 velocity of the wave with respect to the water; pi, 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 ('J), can by partial 

 integration be written 



T S * Cs Wl 7 



Li = 2 J Pi- ^-<ty, 



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

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

 pear. Since now according to equation (1) 



dx ~ dy 



there results 



t K fin 7 



Of, if we put 



which difference is independent of x, we obtain 



and similarly 



ii=|V.fi . (3) 



i 2 =-f>f 2 . : . . (3a) 



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

 given this equation as the first approximation, taking into consideration the strength 

 of the wind. Philos. May., 1871 (4), vol. XL, p. 362, where, moreover, ths influence of 

 capillarity is also considered. 



