PAPER BY PROF. KIRCHHOFF. 133 



where, as also iu the following examples, Tc indicates a positive real 

 fraction, and where the region of oo is bounded by the lines 



//<=0 q>=— oo 



f — Tt ^=+00 



The expression adopted for /( oo) is single valued. The multiple points 

 of Vf(oo)j\oo)—l that do not lie outside the region of go are the points 



<p=-log{l-Jc) f=0 



Cp=-\Og(l + lc) <I* = 7T 



These lie in the boundary of this region, and, therefore, it need not be 

 further bounded by sections. 



The equations of the boundary of the region of oo are also the equa- 

 tions of the bouudary of the region of z. If we assume that for cp= - 

 log (1-1-7;) and ?/< = n we have x = and y = 0, then these equations 

 when developed become the following 



For >p = n and <p < — log (1 -f k) there results 



(A;_e-</>_ y/\k-e-+) % —l)& 



log [fA 



9 



where the root (as also hereafter every root of a positive quantity), is 

 taken to be positive. By these equations the positive half of the axis 

 of x is represented; this is to be taken as a fixed wall; at the initial 

 point of coordinates it merges into the free boundary. For this free 

 boundary, namely, for y=7t and cp > — log (1+&) we have 



-/ 



4> 



(Jc—e-*)d cp 



l<>g(,-K) 



y=- / l—(lc—e-*)* dtp 



1-K) 



Furthermore for tp=0 and cp <— log {1-lc) we have 



x = f(fc+e-*+ J \k+e-*) 2 -l)d<p+a 



J -log 1 -* I * ' 



y=b 

 and for f=0 and cp> — log (1— fe) 



x= I (fe+c-*) dcp+a 



J —log (I—*) 



y=- / Ji-{l'+e-*) 2 dtp+b 



J -log(l-S) 



where a = A ' lo S iJEfc -2- n ^ x ~ ¥ 



b=—2nh 



The first part of the stream line ^=0 which is a straight line paral- 

 lel to the axis of x and extending to the point x=a, y=b, is to be con- 

 sidered as a fixed wall; the second part is to be considered as the free 

 bouudary of the outflowing jet. 



