100 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



currents of uuiform velocity, but in the ueigliborbood of the wavy 

 boundary surface the motion must follow its direction. 



Designate by u and v the comimneuts of tbe velocities of the liuid 

 particles at the point corresponding to tbe rectangular coordinates x 

 and }/; these velocities are by assumption, independent of tbe time, and 

 (for an incompressible Huid whose current is free from vortices) can be 

 presented in the form 



n = — -- 



r = ^-^ ' 

 ^x 



wbere //• is such a function of the coordinate as satisfies tbe differential 

 equation 



'^+'^^"=0 (2) 



Tbe equations 



?/-=const. 



are in this case, as is well known, tbe streamlines of the tluid. Tbe 

 boundary line of both tluids must be such a stream-line, and we will 

 give it for both sides the value 



cS = and ^■2=^- 



Tbe above overscored letters will, in what follows, always indicate 

 values on the boundary surface. 



The first boundary condition that we bave to satisfy is therefore that, 

 wbeii we express (.S and c'-o as functions of x and y. then tbe two equa- 

 tions 



?'.=o=y^2 (2«) 



sball admit of an accordant solution. 



Tbe second boundary condition is that the pressure at tbe bounding 

 surface sball be the same on both sides, or 



lh=lh (2i) 



Now, under tbe adopted assumptions and when s is the density of 

 tbe tiuid and C is a constant, we bave 



Therefore the equation {2b) can be written : 



Const. = (.,-.,)^i+i.,(|||/-^J;;^y .... (3) 



