PAPER BY PROF. KIRCHHOFF, 137 



Finally we assume that go and z disappear simultaneously. 

 At the boundary of the region of z we have, first the line for which 

 ^'=0, and <p>0. This line is composed of the following portions : 



For ^=+0, and 0<(p<— log (1-F) we have, 



x 



-/ (tt^^+Vi^- 1 y* 



y=0 

 For if>=— and 0<<p<— log (1— ¥) we have, 



These equations represent a portion of the axis of x that is to be 

 assumed as the fixed partition. Adjoining this fixed partition there 

 comes as the free boundary of the moving fluid the line for which 



y=+0, ^>_log(l-fc 2 ), 

 therefore 



dx = h d V_ = _ f~T ~P~ 



d<p Vl—e-^ , d<p v i_ <?-* ' 



and also the line for which 



,/-=_0, <p>_log(l-£ 2 ), 

 whence 



— = _ ft ^ = _ /T~ ~^~ 



d<p — Vl— c-*' <*<P v 1— e-* " 



The remaining boundaries of the region of 2 are the lines 



f= — 7r, i/: = + 7t, cp=— go, <p=-foo. 

 For 7p= — 7r we have, 



For //'=4-7r we have, 



d.r_ fc dy_ l~ ft 2 



dcp~ 1+e-*' <^ — — \ l+e-* ' 



These two stream, lines are free boundaries throughout their whole 



extent. 



,, . <7z . 



For q)=—cc, we have^— ==— * ; 



for <p=+co, and >/'<0, we have ■=- -= — A; — i Vl— ¥ 



and for <p=+co, and >/•>(), we have _=fc_« \Zi_F 



