PAPER BY PROF. KIRCHHOFF. ioo 



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

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



tp=0, q)= — -^ ; 



The expression adopted for /(a;) is single value. The multiple points 

 of Vf\a))Y{oj)—l that do not lie outsidethe region of go are the points 



^=-log(l-A-), ^-=0; 

 (p=-log (1+A;), ip^Tt. 

 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 co are also the equa- 

 tions of the boundary of the region of z. If we assume that for (p= - 

 log (1 -|- A-) and //' = tt we have x = and y = 0, then these equations 

 when developed become the following 

 For tp^Tt and cp <^ — log (1 + A) there results 



/•* 



y=0 and .r= j (/,-_e-*_ ^(/^_e-<|.)2_l) d cp 



t/— Ice iH-i) 



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 fieii 

 boundary, namely, for H' = 7r and cp y — log (l + A) we have 



(A- 



_ 1,,„ ( J.!- 



e-*)fZ q) 



- leg ( 1+i) 



y=- / ^l-{l--e-'^]^ dcp 

 Furthermore for i/-=0 and q) <— log (1 - A) we have 



J -log i 1 -* ) ^ / 



y=h 

 and for ip=i) and ^> —log (1— A) 



x= I (fc+e-"^) dcp+ci 



1/ -log(l-i) 



y=- / Jl-{-k-\-e-^Ydcp+b 



where « = A- log +- -2— tt ^iTTF 



h=-2 7tk 

 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 

 boundary of the outflowing jet. 



