64 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



themselves functious of x and y. But inversely we can consider and 

 develops {- yi as a function of ©+///. In problems relative to cur- 

 rents between two stationary walls, ip is constant along the boundaries, 

 and therefore if 9? and ?/■ are presented as rectangular coordinates in a 

 plane, then in a strip of this plane bounded by two parallel straight 

 lines, ip = Co and ?/' = Ci, the function x + yi is to be so taken that on the 

 edge it corresponds to the equation of the wall, but in the interior it 

 assumes a given variability. 

 A case of this kind occurs when we put 



x + yi = Alcj,+ >l-+e''+'^'\ = . (2) 



or 



<b 

 X = A (p + Ae cos if- 



y=z A //• -\- Ac sin 1/' 



For the value ^' = ± ;r we have y constant and x = Aqj — A€'. 



When q? varies from — x to + cc the value of x changes at the same 

 time from -co to —A, and then again back to —go. 



The stream Hues x=±7r correspond tb us to a current along two 

 straight walls, for Avliich y = ± Att and x varies between — cc and — A. 



Therefore when we consider '/' as the expression of the stream curve 

 the equation (2) corresponds to the tiow out into endless space from a 

 canal bounded by two parallel planes. On the border of the canal 

 however where x= — A and ?/ = i A ;r and where further, cp = and 

 ^= ± ;r, we have 



dcpj V dcp 



therefore 



0, 



C.i)'Hf^ 



Electricity and heat ilow in this manner, but liquids must tear asunder. 

 If from the border of the canal there extend stationary dividing dis- 

 continuous lines that are of course prolongations of the stream lines 

 ip= ^ 7t that follow along the wall and if outside of these discontinuous 

 lines that limit the flowing fluid there is perfect quiet, then must the 

 pressure be the same on both sides of these dividing lines. That is to 

 say, along that portion of the line ip= :L tt which corresponds to the 

 free dividing line, in accordance with the equation (1&), we must have 



In order now, in the solution of this modified problem, to retain the 

 fundamental idea of the motion expressed in equation (2), we will add 



