STEADY MOTIONS OF EARTH S ATMOSPHERE ERMAN 35 



Therefore, we now have 



p p 1 F. 



p = = or - = — 



F (x, y, z) F. p p 



and the left-hand portions of the equations (I) become respectively 



1 dp = F dp =F dlogp =x 

 p dx p dx dx 



1 dp = F dp = p dlogp =y 

 p dy p dy dy 



I dp = Fty = F d log p _ z 



p dz p dz dz 



where log indicates the natural logarithm. 



Since it is known that the components x, y, z, of the resultant 

 of gravity and centrifugal force, as also the components u, v, w 

 of the velocity of a fluid particle, depend only on x, y, z or on the 

 variable coordinates by which we express the location of any point 

 on the sphere, therefore, as is well known, there are two functions 

 of x, y, z, which I will designate by V and <p respectively 5 which are 

 determined by the relations 



x = dV . V = dV • Z dV 



and 



dx dy dz 



d<p do do 



— ; v = - ; w = — 

 dx dy dz 



If these values are substituted in equation (I) and the sum is 

 taken after multiplying the first, second, and third respectively by 

 dx, dy, dz, and if we recall that 



5 These functions are now generally called the force potential, V, and 

 the velocity potential, f, or the potential function for the external forces and 

 the potential function for the resulting velocity. If such functions actually 

 exist there can be no discontinuous whirls, and if the whirls exist then there 

 can be no such functions. — C. A. 



