where [real] indicates that the integral is taken along the real axis. Thus, in general, n 

 factors in dz/dt produce a polygon with 71 + 1 vertices and sides, and with i = ■» at one vertex. 

 If a, + a, + • • • **n ~ ^' however, a„^ j = 0, so that the first and last segments of the 

 broken line coalesce into a single straight line. In this case the polygon has only n actual 

 vertices and sides, and < = 00 occurs somewhere on one side; the total length of this side is 



a. 00 



{ [real] ^ dt ^ \ [real] 



dz 

 dt ^^ 



This case may be regarded as a degenerate one in which the exterior angle at the (n + 1) st 

 vertex is zero. 



Thus it has been shown that the real axis of t is transformed into a finite closed polygon. 

 It remains then to show that the arbitrary constants in the transformation can be chosen so as 

 to fit an arbitrarily chosen polygon on the 2-plane. 



The general expression for z will be 



-a, - a-, -«, 



K (t-a^) (t-a^) ...{t-a^ dt + L 



Now changing the integration constant L merely translates the polygon on the 2-plane; changing 

 \K\ stretches all of its sides in a certain ratio, and changing amp K rotates it about the point 

 z = L. By adjusting K and L, therefore, one side of the transformed polygon can always be 

 made to coincide with one side of the given polygon. The two polygons will then coincide 

 completely provided they have the same shape. Th'e necessary similarity can be secured in 

 either of two alternative ways. 



1. For a polygon of m sides, m-1 factors may be employed in the expression for dz/dt, 

 with a , a . , a^_. made equal to m-1 external angles of the given polygon taken in 

 order, each divided by n. The external angles then come out correct. For the lengtlis of the 

 sides, m integrals are obtained, two of them extending to t = + <x. Elimination of the factor 

 l( from these integrals leaves m-1 ratios between them. By a suitable choice of a , a^- • • 

 a^_., these ratios can be made equal to the m-1 ratios of the lengths of the sides of the 

 given polygon to the length of a chosen side. These latter ratios cannot all be independent, 

 however; for the last two sides, whose directions are already fixed, will automatically come 

 into the correct ratio when the other ratios have been adjusted. In F'igure 34, for example, 

 A^ A^ and ,4^ A^ are fixed in length when their directions have been assigned and when the 

 sides /4j A^ and A^ 4, have been constructed in the proper ratio to A^ A^. Hence two of the 

 a'a can be chosen arbitrarily, the remaining a's being then chosen so as to give correct values 

 to 77! -3 of the ratios of the sides. In practice, it is usually most convenient to determine K 

 and L by substituting the values of z at two corners. 



63 



