STEADY MOTIONS OF EARTHS ATMOSPHERE ERMAN 37 



dlogp 1 BV blogF 1 \\dxj \byj \bz I / 



bx F dx dx 2F dx 



, •{(ZHtHi)! 



b log p 1 bV b log F 

 dy F dy by 2F 3;y 



Olog/j 1 3F 3logF 1 {\dx/ \byj \bz 



(A*) 



bz F bz bz 2F 



and we can, therefore, by simple substitution of A* in B construct 

 an equation in which, in addition to the partial differential quotients 

 of the first and second order of the function <p — <p(x y z) there enter 

 only the known functions F = F (x, y, z) and V — V (x, y, z) and 

 their first differential coefficients. 



If x, y and z are replaced by the angular coordinates of any point 

 of the atmosphere so that 



x = r cos X V\ — p. 2 = r cos /? cos X 



y — r sin X Vl — fi 2 = r cos ft sin X 



z = r n = r sin /? 



where r is the distance from the center of the earth to any point 

 in the atmosphere; X is the longitude of this point; /? is the latitude 

 and jj. is the sine of the latitude, then, as is well known, we have 7 



y = (r * 2) -Hi)-' 2 - 2 > 2 -^ 2 



where for the surface of the earth and at the equator, we have 



r =R 



y = acceleration due to the attraction of the earth 



T 



a = acceleration due to centrifugal force = ~i— • 



289 



7 This expression assumes that gravity and centrifugal force are the only 

 external forces and thus ignores viscosity or internal friction and the resist- 

 ance of the earth's surface and the attractions of sun and moon. 



