300 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



Again, we make the transformation 

 and derive the surface 



Here 



and for the value £ = we have the roots 



? = * j I = — **• 



Let 



$3=1 — i, 

 and we have the surface 



L 2 + 2*& - e - ?V& - i?V = 0, 



1 + h ^ / ?t ? 2 +4| =2 -4. 



whence 



£ 3 = 



2 



In a similar way, from the other root, 



(c) 



(d) 



In (c) and (d), for the radical is taken only that branch which becomes 

 + 2 i for zero values of the arguments, and the function is seen to be 

 analytic for sufficiently small values of q when 



CI 



'/ 



< i - « i ; 



and similarly when 

 V 



\v\ = 



> 1 + e. 



Thus, in the ^-plane, we have 

 by the formulas (a), (b), (c), (d) 

 covered two, small circles about 

 the points 1 and — 1 corre- 

 sponding to developments (a) 

 and (b), and all of the region 

 outside of a circle of radius 

 (1 + e), corresponding to devel- 

 opments (c) and (d). We must 

 now obtain further formulas 

 so as to till up the remaining unshaded region. 



