PAPER BY PROF. KIRCHHOFF. 135 



(II.) As a second example the case where 



f{c.>)=k+ - 

 V cj 



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

 tions. 



In order to make f{co) a single- valued function we draw a section 

 from the jioint oj=0, for which section ^'=0 and ^>0 and assume that 

 for ^= + and ?/'=+0 the real part of -/ci^ is positive. The cusp points 



of the curve ^/Jl^i!^)Y{GJ)^^ilve the points for which cj=0 , =1— ^^ 



vcy 



-7— = — (l+A-); therefore they all lie on the section already drawn 



therefore do not require the making of a new section. As concerns the 

 sign of V;'{oL)}f{Go) — i it must be so determined according to the 

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

 for ^= + 0, and //' = -|-0. Finally it is assumed that gj and s disappear 

 simultaneously. 



The line for which ?/=0, and (p>0, is the boundary of the region of 

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



from each other. For //'== + 0, and 0<ot< i we have 



(I-Af 



Then again for '/=— 0, and 0<fp< -^ 



' (1— A-) 



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

 adopted as the fixed wall. If we use the relation 



/V(>+7=z)-i^'''^= 



Iirfc2 ^(AV(p-fl)--^+ ^j^_y,,^j arf sinni-c2);/<^_fcj 



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

 1 + A--A'^ 1 /;r , . ,\ 



•^=2(i^£l)(i:iF)+ (r:^e C2 + '^^^ ^^" 'V 



and 



1-A--A2 





(1+^(1 -A^) -(T^.(g- "'■"«'"*) 



