300 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



Again, we make the transformation 



and derive the surface 



Here 



e+ i-i^-H^v = o. 



and for the value ^ = we have the roots 



l=i, I = — I. 



Let 



^3 = ^ — «■ > 



and we have the surface 



$,' + 2t^3 - ? - Cvis - i^'v = 0, 



whence 



i, = ^^2L-li + ^VcW + H'-^- 



In a similar way, from the other root, 



$, = ^''^'^^^' -h^CW + H'-^' 



(c) 



(d) 



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

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

 analytic for sufficiently small values of rj when 



C = 



V 



< 1 - f 1 ; 



aud similarly when 



l'?l = 



> 1 + e. 



Thus, in the ry-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 fill up the remaining unshaded region. 



