118 THE MECHANICS OF THE EARTh's ATMOSPHERE. 



of tlie individual forms of tlie wavt's. At least one cusp value occurs 

 lor a plane surface. 



Only this much can be at once seen, that when an absolute minimum 

 exists there must be a transition leading from this to the infinite nega- 

 tive value of ((?— i), which transition at first begins with an ascending 

 value and then again diminishes. There mnst then be a lowest value on 

 the transition curve between the ascending and the descending values 

 that corresponds to a maximo-miuimum (absolute minimum ) of the quan- 

 tity (^— i), therefore also to a stationary form of wave, but such an one 

 as corresponds to an unstable equilibrium, and which is on the point of 

 becoming a breaker. 



If such a minimum exists, then for it any variation in the form of the 

 "wave that makes <^ increase will make L increase by the same amount. 

 The same is true of the cusp value when we consider such waves as 

 form trough-lines. But if we increase the values of pi and 1^25 that is 

 to say, if we increase the velocity of the wind and the rate of i)roi)aga- 

 tion of the waves through water, then the partial differential coeflicient 

 of X will be greater at both places and the two limiting values must 

 approach each otlier and finally coincide, whereby the absolute mini- 

 mum ceases to exist and the equilibrium becomes unstable. Hence it 

 is to be concluded that with increasing rate of flow, stationary waves 

 of a given wave-length will finally become impossible. 



Necessary formation ofhrealers ichen the velocity is excessive, — Tliat,for 

 a constant definite value of the wave-length, minima of the function 

 (0 — L) are no longer possible for large values of pi and p2 exceeding a 

 certain definite amount, can easily be shown as follows: We compute 

 the values of ij and X2 under the assumption that ).'>i=po = l, for any 

 arbitrarily chosen form of wave and then for an arbitrarily chosen value 

 of d<P seek the two variations of the curve which respectively^ make 6L1 

 and 6L2 to become maxima. 



Among the possible variations of the form of the wave that give 

 positive values of 6^ are those that give higher summits and lower 

 troughs for the wave. Since the upper fluid has the greatest [least ? — 

 C. A.] section above the summits of the waves, but the smallest [great- 

 est"? — C. A.] section above their valleys therefore above the summits a 

 greater velocity of flow must prevail tban above the valleys, that is to 



say the value of -^w wiHst be greater absolutely on the summits than 



in the troughs. Hence follows from equation (2e) that when we raise 

 the summits and depress the valleys we obtain not only positive values 

 of d<P but also positive values of dLi and 6L2. Consequently the de- 

 sired maximum values of the two quantities SLi and 6L2, that belong 

 to the prescribed positive values of d'P are necessarily positive, and for 



a finite altitude of the wave the ratio v:i-as also -r-r- must necessarily 

 be finite. 



