where n and c are positive real numbers. The points z = - c now correspond to z' =- nc, and 

 at these points, in general, comformality fails. 

 Writing, as illustrated in Figure 141, 



z - c = r.e , z + c = r^e , z - nc = r.e ^, z -\- nc = Tr,e "', 



where -■n^B^^rr,-7j^d^^n, 

 it follows that 



};-d;^n{d^-d^). 



L88b] 



Figure 141 - Illustration for a circular arc A, B or C. See Section 88. 



This shows that any circular arc joining z = - c, along which 6^ - 9^ hsis a constant value, 

 transforms into one joining z' - - nc. The tangent to the z arc makes an external angle 

 y = 0j - ^2 ^>*'h i*^ chord produced beyond 2 = c, or with the positive a;-axis; the tangent to the 

 2 'arc makes a similar angle y'with its chord where 



y'=7iy. [88c] 



Here - n ^ y ^ n. The respective radii of the arcs, which subtend angles 2?? - 2y or 2n- - 2y' 

 at their centers, are R = c / |sin y|, S'= c/ |sin y'|. 



To solve for a;' and y', where x' -¥ iy' = z', write 



3 + c '2 



4 = In = A - z/x, X = In — , n = 6. - 



z - c Tj 



[88d,e,f] 



Then (z'-t- nc)/(z'— nc) = <?"^and, solving for z', 



z' - nc coth 



nC 



211 



