Prof. Helmholtz on Discontinuous Movements of Fluids. 343 



From equation (1) we get as integral equation that the value 

 <f> -\-^i is a function of x + yi (where i= V — 1). The solutions 

 hitherto found generally express cf> and ^as a sum of terms 

 which are themselves functions of x and y. But, inversely, we 

 may regard x-\-yi as a function of cj)-\-yjri and develope. In 

 problems dealing with currents between two rigid walls, yjr is 

 constant along the borders ; so that if <£ and i/r be laid down as 

 rectangular coordinates in a plane, we have, in a strip of this 

 plane bounded by two parallel straight lines ty = c and ^ = Cj, 

 to seek the function x = yi so that it may correspond at the edge 

 to the equation of the wall, and in the interior assume the given 

 discontinuities. 



A case of this kind is when we put 



x + ?ji = A{(j) + ^i-)-eV + * i }, .... (2) 

 or 



x = A<f> + AeV cos ^jr, 



y = Ayjr + Ae sin -yjr : 



for the value -v/r= + tt, y is constant, and 



x = A(j>— Ae®. 



If <j> varies from -co to -fee , x changes at the same time 

 from — x to — A, and then back again to — go . The current- 

 curves yjr= +7r correspond, therefore, to the current along two 

 straight walls, for which ?/=±A7r, and x varies between — oo 

 and —A. 



If, therefore, we use -v^ to express the current-curves, equation 

 (2) corresponds to a current which flows into infinite space from 

 a canal bordered by two parallel planes. At the edge of the 

 canal, however, where x= — A and y= ±A7r, and, further, 

 where </> = and yjr= +ir } 



m* (D*-» 



dy t 



Electricity and heat may flow in this manner ; liquids would be 

 torn asunder. 



If from the edge of the canal there should proceed stationary 

 lines of division, which of course would be continuations of the 

 current-lines yfr=+ir passing along the wall, and if exteriorly 

 to these lines of division which border the current fluid there 

 should be rest, then the pressure must be the same on both sides 

 of the lines of dvision. That is, along those portions of the 



