PAPER BY PROF. KIRCHHOFF. 137 



Finally we assume that co and c disappear simultaneously. 

 At the bouudary of the region of ~ we have, first the line for which 

 ^•=0, and ^>0. This line is composed of the following* portions: 



For //■ = + 0, and 0<(^<— log [i—k'') we have, 



*/' A,- 



.rC vr^F?+Vi^.,-i y^p 



For //•=— and 0<(^<— log (I — A') we have, 





y=o 



These equations represeut a portion of the axis of ,v that is to be 

 assumed as the fixed partition. Adjoining this fixed partitiou there 

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



./'=+0, cp> -log {l-Jc% 

 therefore 



drp -/i_e-*' dq) V l — e- 



■ e-'t' 

 and also the line for which 



//'=-0, cp>-\0g(i-k'), 



whence 



dx k dy 



= - /i ^ 



dq)~ A/i_e--^ ^ dq)' 

 The remaining boundaries of the region of z are the Hues 



>f- = — 7r, y- = 4-,T, (7>=— Gc, ^= + 00. 

 For if:= — 7T we have, 



dx_ k _ dij 



d(p~ sJiJ^e^V dqi V --j^g_,|, 



dy _ I p — 



For i/- = -{-Tr we have, 



These two stream lines are free boundaries throughout their whole 

 extent. 



T-i 1 dz 



lor m=—y: we have-r- = — * ; 

 ' doo ^ 



dz 

 for ^=+Go, and '/■<0, wehave t— =:_A;-i vT^^^ 



and for (p= + y3, and '/■>0, we have ^^ =k — i y/Y—k^ 



doj 



