344 Prof. Helmholtz on Discontinuous Movements of Fluids. 



lines i/r=+7r which correspond to the independent divisional 

 lines we must have, according to (\b), 



In order now to maintain the chief features of the motion 

 given in equation (2), let us add to the above expression for 

 oc + yi another term a + ri, which is also a function of </> + yfri. 

 We have then 



%=A(f> + Ae® cos yjr + o; ~\ ,„ n 



y = Aty + Ae® sin tJt + t; J 



and we must determine a + ri so that, along the free portion of 

 the divisional surface i/r= ±ir, we may have 



( A - Ae?+ S) 2+ © 2=const - 



This condition is fulfilled if, in the same place, we make 



-jv =0, or cr= const., (3 b) 



dcp 



and 



dr 



d ,=±A\/2eV-e 2< ? (3c) 



Since yjr is constant along the wall, we can integrate the last 

 equation in regard to <f>, and convert the integral into a function 

 of (/> + i/^' by putting </> + i(-^r -f it) everywhere in place of <f>. 

 We thus, after properly determining the constants of integration, 



get 



o- + T*=Ai| ^_2 e cp+^_ e 2cp+2^ + 2 ar c S in[-^|^ ((p+ ^ ) ] |.(3d) 



The points of divergence of this expression are where 

 g<p+*t__2, that is, where ^= + (2a + 1)tt and <£ = log2; so 

 that none of them lie in the interval from -\^= -f 7r to i/r= — it. 

 The function a + ri is here continuous. 

 Along the wall 



(r+Ti= ±Ai^ ^2^—6^-2 arc sin |_77^ e ^J J ' 



If < x|r^log2 the whole value is purely imaginary, so that cr = 0, 



dr 

 while Tj assumes the value given in (3 c). This portion of 



the lines yjr=±7T, therefore, corresponds to the free portion of 

 the stream. 



