110 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



stream with the plane bounding surface. But the volumetric extension 

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

 is greater than 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 Z which first oc- 

 curred in the computation by considering the terms with r : ' in the val- 

 ues of x and y, give a basis for the computation of the difference of 

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

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

 follows : 



E-^A: ^=&\ [5£ 2 -2* 2 ]+^ [^-15^C 2 -|C 4 ] 



1 =a z - 



2 7T<J (s— s ) 4 



or 



E, 



'27T</(.V 2 -S,) 



7cos 2 £ r4 [5-2cos 2 f][cos 2 -£J _ 1 C 4r 15 .0845^cos 2 fl.fcos 2 £-fa.0845] 

 "US - " - cos 2 f-# 48- l ' 



144 cos 2 f— f 



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

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

 computation. For E = 0, we fiud the value cos-'f = 0.G75148, 

 which is not very far from the limit of couvergency or cos£ 2 = 0.G7264. 



Corresponding to E = we find 



D = 0.740333 

 %< = 0.1717613 



C = 0.6899 



z = 0.56686 

 E = 0.20404 x A 

 e h = 2.52000 



Since these are the waves that can be immediately produced by a 

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

 of the computation quoted in Article 6, whereas the values for the 

 lowest waves are found when we assume for cos 2 £ the upper limit of its 

 values, namely, 0.68615. 



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 2 e > 0.675148 is smaller than that of the recti- 

 linear How of both media with the same velocities. The difference 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 boundary value. 



