PAPER BY PROP. KIRCHHOFF. 131 



The partial differential equation for cp is satisfied if we have 

 z=x+iy w-.p+i^ 



where i= V — l, and &> can be any function of z. Therefore 



the equation of any curve of flow or stream line is f =constant, and we 



have 



dx 

 ?<p d<p 





Jcp dq> 



\JcpJ + VcW 

 ( d<p \*±( ?(f) \ 2 1 



if we assume that x and y on the right-hand side of these equations can 

 be represented as functions of cp and ij\ Therefore the conditions for 

 a free boundary of the jet are that for it </==constant, and 





i. 



The problem is therefore to express go as such a function of zaswill 

 satisfy these conditions. 

 To this end we put 



aud select the function /(<y) so that it is real for a certain value of xp 

 and for a certain range of cp, and so that it lies between the limits —1 

 aud +1. For this value of ip and for this range of cp we have 



whence 



£-■««>. f£-VWl«Wt"> 



£X£)-' 



that is to say, the stream line corresponding to the value of tp can form 

 a free boundary to the moving liquid in that portiou which corresponds 

 to the range of cp. If there are many values of >p for which /(a?) has 

 the described property then all the stream lines that correspond to 

 these values can be free boundaries. 

 In general co is defined by the equation above given for 



dz 

 doo 



as a many- valued function of z for any definite assumption as to /(<»). 

 Let the region of z, that is to say the space filled with the moving liquid, 



