100 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



currents of uniform velocity, but in the neighborhood of the wavy 

 boundary surface the motion must follow its direction. 



Designate by u and v the components of the velocities of the fluid 

 particles at the point corresponding to the rectangular coordinates x 

 and y; these velocities are by assumption, independent of the time, and 

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

 presented in the form 



11 — — / 



dy 



where ip is such a function of the coordinate as satisfies the differential 

 equation 



^+^"-0 (2) 



The equations 



ij-= const 



are in this case, as is well known, the streamlines of the fluid. The 

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

 give it for both sides the value 



V'i = 



=0 and c''2=<>- 



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

 values on the boundary surface. 



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

 when we express fa and 4> % as functions of x and y. then the two equa- 

 tions 



^=0=^ 2 (2a) 



shall admit of an accordant solution. 



The second boundary coudition is that the pressure at the bounding 

 surface shall be the same on both sides, or 



Pi=P-2 (2b) 



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

 the fluid and C is a constant, we have 



Therefore the equation (26) can be written : 



Const, = (. Sl - S2 )^^ + ^^0'_^^y .... (3) 



