PAPER BY PROF. KIRCHHOFP. 135 



(II.) As a second example the case where 



will be treated and the region of go stretches indefinitely far in all direc- 

 tions. 



In order to make/(&?) a single- valued function we draw a section 

 from the point oo=0, for which section ?/=0 and <p>0 and assume that 

 for y>= + and '/=+0 the real part of V o is positive. The cusp points 



of the curve y/Tfoj) /'(<»)— I are the points for which go=0 , — _=1— Jc, 



1 



—. — = _(i + A:); therefore they all lie on the section already drawn 



VIP 



therefore do not require the making of anew section. As concerns the 

 sign of Vf (&))/( go)— I it must be so determined according to the 

 adopted rules that the real part of this radical quantity shall be positive 

 for <£>= + (), and i/- = -\-0. Finally it is assumed that go and z disappear 

 simultaneously. 



The line for which >/•=(), and <p>0, is the bouudary of the region of 

 z. This line is composed of many parts which are to be distinguished 



from each other. For ?/>=+0, and 0<<p< jy^hvi we bave 



y=o. 



Then again for >/•=— 0, and Q<V<rr^\2 



These equations represent a part of the axis of x which is to be 

 adopted as the fixed wall. If we use the relation 



JV( 



1 \ 2 

 v cp ' 



we find for the end of this part (of the axis of x) the expression 



l+fc-fc 2 i_ /*. arc sin A 



and 



