l86 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 5 1 



u at the distance z by the formula 



u = u o( h _ z ) = u - U °z (1) 



h \ / h 



The internal friction per unit of surface which we denote by F 

 will be equal to a coefficient K multiplied by the rate of increase of 

 velocity and consequently 



F = K~° (2) 



h 



The plane A B moves with the velocity V and the air along A B 

 moves with the velocity u ; the resistance between the air and the 

 plane .4 B is proportional to the difference V — u and to the coeffi- 

 cient of friction f between the air and the plane; consequently we 

 can write 



F = / (V - u ) (3) 



From these equations we find 



*-,-& <« 



h 



In the preceding case the pressure has been supposed to be con- 

 stant, We shall now consider the case where the horizontal cur- 

 rent of air has a gradient ; the horizontal velocity u depends solely 

 on the distance, z and the vertical velocity is zero. The increase of 



du 

 the horizontal velocity per unit of vertical distance is j- and the 



du 

 internal friction is equal to K —r- . Considering a parallelopipedon 



whose thickness is dz and whose face is a unit of area, the result of 



du 

 the frictions of the two faces will be d(K — ) and the mass of the 



element will be pdz. The force per unit of mass resulting from the inter- 

 im d 2 u 



nal friction will therefore be — -7^ and this force acts in the same 



p dz 2 



direction as the force of the gradient. The equation of equilibrium 



is 



V G= _K<Pu (g) 



p p dz 2 



