PAPER BY PROF. KIRCHHOFF. 131 



The partial differential equation for qj is satisfied if we have 



where i= -/ITr, and o; can be auy function of ^. Therefore 



the equation of any curve of flow or stream line is //•=coustant, and we 



have 



Jx 



dq> d(p 





njp\\njp\'^ 



\»J^\3yJ C^^\',f']lX 



(.%y-m 



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

 be represented as functions of q) and //•. Therefore the conditions for 

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



crjvc;^-- 



The problem is therefore to express co as such a function of ^aswill 

 satisfy these conditions. 

 To this end we put 



(Iz 



and select the function f{co) so that it is real for a certain value of ip 

 and fur a certain range of <p, and so that it lies between the limits —1 

 and +1. For this value of //' and for this range of cp we have 



whence 



that is to say, the stream line corresponding to the value of ?/' can form 

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

 to the range of q). If there are many values of if: for which /(cy) has 

 the described property then all the stream lines that correspond to 

 these values can be free boundaries. 

 In general gj is defined by the equation above given for 



doo 

 as a many-valued function of z for any definite assumption as to f{co). 

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



