582 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 5 1 



The computation of the available kinetic energy for this case 

 supplements the computation for air whose entropy is a function 

 of length and altitude. The angular inclination to the horizon of 

 the layers of equal density is 



♦ t° x h 

 a = arc tan — . — 



\°z l 



If o x >o z then the layer that starts from the edge where % = o'and 



z = h will intersect the bottom of the trough. 



The analysis must be executed separately for the three re- 



% z 



gions in which <p = o x - + a z is between o and a z ; or between 



/ h 



a z and a x \ or between o x and a x + o z . The first and last of these 



regions are triangles in the xz plane, the second is a parallelogram. 



The evaluation of ~P a and ~P e for the separate regions is rather 



tedious. Eventually we find 



If a x < o z then the above described layer (that starts from the edge 

 for which x = u, z = h) will intersect the opposite vertical wall of 

 the trough. For this case we have 



a 



In this case also the available kinetic energy is independent of the 

 length of the trough. 



If a z = o [then a = 90 and the layers are vertical columns as in 

 case B and] the former of these two equations becomes 



V 2 1 1 — — ) = 2)1 — which is identical with the value of V com- 



V 2/ 6 



puted in case B when o x is a small fraction. 



If a z = a x then the layers are parallel to the diagonal surface of 

 the trough. For this case the two equations agree in giving 



gha 



therefore, if a is a small fraction, the available kinetic energy is 

 about one-fourth of that which we found for the same value of a 

 in case (^4) or two-fifths of the analogous quantity in case (B). 



