192 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



a rather complicated value for the angular velocity X. I have intro- 

 duced a simplified expression for this in that, while retaining the 

 dependence upon the polar distance 0, as there given, I have tempo- 

 rarily introduced a constant average value instead of the dependence 

 upon the distance above the surface of the earth. According to this, 

 one can put 



x =xi cos 2 0-x-z (3) 



or with a slight difference 



X 



=& | Xiz 2 -X>r 2 J (4) 



In these equations Xi and j 2 a-re considered as constants. Therefore, 

 as before found, the movement of rotation of the air in higher latitudes 

 is positive, that is to say, has the same sign as the axial rotation of the 

 earth. For a specific latitude the average value is 0, and at the equa- 

 tor the movement has the opposite sign. 



Further computation shows that the relative angular velocity j is 

 small in comparison with that of the earth £, so that the simpler equa- 

 tions to be solved are as follows : 



c 2 j~ = 2e xx + h Au 3 



(5) 



In solving these we first determine a function g that is of such form 



as to satisfy the conditions ?a n ^A n „ 



~ = 2exx,^eXy. 



These conditions give 



^ S v „2 Kir* \ 



*=%\ 



Xx* - % rZ j (6) 



Furthermore we put 



*=%>—$>--£ + * !7) 



where L and M are two new functions of x, y, and z, we can then write 

 the system of equations as follows : 



C 2p = S + K d_ {AL) 



d% dx d% 



J J/ 3 _ d% „ I 



C 2 ^- 



+ x L —{AL) 



dy dy dy 



