82 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 51 



where q is a constant and has very nearly the value of 1/8000 if 

 v, the difference in altitude, be expressed in meters. In this case the 

 following also holds good: 



£ 



and, consequently, 



1 de 



e dy 



hence the differential equation for y> becomes 



'*-«■£ (2) 



ay 



A solution of this differential equation that satisfies the assump- 

 tions 5 and 6, is given by the expression 



<p = a (x — b cos m x. e~ ny ) (3) 



in which the following relation exists between the constants m and 



m 2 _ n 2 _ g n . 



n = — - + r, where r = V m 2 I o 2 4. I 



2 j 



In order to ascertain what profile or configuration of the ground 

 corresponds to the current determined by this velocity potential, 

 we must look for the lines of flow; for one of these must certainly 

 agree with the profile curve. The differential equation of the stream 

 lines reads as follows: 



dy.dx = ' : ' = abn cos mx.e~ n " : a (1 + bm sin nix e " !l \ 

 dy ox 



The integration of this equation gives 



111 



e~ ny . sin mx = - + Be q,J (5) 



bqn 



wherein B represents the parameter of the stream lines. 



If we assume that the curve of the profile of the surface passes 

 through the points x = o and y = o, then for these values B = 



