110 THE MECHANICS OF THE EARTHS ATMOSPHERE. 



stream with the plane bouudmg surface. But the volumetric exteusioii 

 of the part with diminished living force, that fills the broader channel, 

 is greater thau the volume of increased velocity in the narrow channel 

 Therefore in the sum total the living force of the diminished portion 

 prevails. 



^Nevertheless only the terms of the fourth degree in C which first oc- 

 curred in the computation by considering the terms witli T^ in the val- 

 ues of X and y, give a basis for the computation of the differeuce of 

 energy. This difference, as couiputed for one wave-length according to 

 my calculation in the class of waves discussed in Section Yii, is as 

 follows : 



or 





E. 



Tcos^g , [5-2cos^.][cosV-ii,J ^ lr4ri5.0S45-eos^.l.[cos^-f+0.0S43] 

 J 44 • cos^f— t 48" ^ J L ^ J 



In this the Cl is the only factor that changes rapidly for small changes 

 of cos f, a circumstance that very materially lightens the numerical 

 computation. For jE = 0, we find the value cos-£ = 0.075148, 

 which is not verj^ far from the limit of convergency or cos£- = 0.672C4. 



Corresponding to E = we find 



e = 0.740333 



^;i = 0.1717(513 



r = 0.G899 



z = O.oOCSfj 

 77 = 0.20404 X A 

 e" = 2.52000 



Since these are the waves that can be immediately x)roduced by a 

 constant wind, therefore these are the values that lie at the foundation 

 of the computation quoted in Article G, whereas the values for the 

 lowest waves are found when we assume for cos^s the upper limit of its 

 values, namely, 0.08015. 



Theory shows, moreover, as also the above numerical example, that 

 the waves of this form for large values of cos e and for the same mate- 

 rial and same strength of wind have greater wave-lengths ; that, how- 

 ever, their altitudes form a smaller fraction of the wave-length, and that 

 their energy when cos^f >0.e75148 is smaller than that of the recti- 

 linear fiow of both media with the same velocities. The differeuce of 

 energy is zero for very low waves ; it is negative when we pass to rel- 

 atively high waves; it reaches a maximum, then diminishes, and is 

 again zero for the given l)oundary value. 



