Leathi^m — Periodic Conformat Curve-Factors and Corner-Factors. 45 



corners. These, if they do not form a closed polygon, have at any rate a 

 space-periodicity which is generally of a circular type, but may be linear. 

 The conformality of the representation of the region •v/r>0 is readily verified. 

 8. Conformed representation of the space outside a triangle or closed polygon. — 

 Let the angles of a triangle in the s plane be A, B, G, and let the values 

 a, /3, 7 be assigned to ^ at the corners. By the previous article it appears 

 that the conformal transformation of the region outside the triangle is deter- 

 mined by the formula 



A B^ (^ 



where K is a constant, and A > 7 - a > /3 - «> 0. The obvious periodicity of 

 the expression on the right-hand side of the formula is for a linear period 2A ; 

 but when it is noticed (i) that the modulus has a period X, (ii) that the decrease 

 of vector angle as vj passes through real values from say w,) to W(, + A is 



TT - A + IT - B + TT - C = 'Itt, 



it becomes clear that the expression is periodic with linear period A. 



In general, though the transformation (22) gives a periodic dz/dw, there is 

 no reason why it should give a periodic z. Usually the value of z for w = a + A 

 will be different from that for vj = «, and the boundary will be a continuous 

 recurring rectilineal pattern having the kind of periodicity that would be got 

 by printing from a rolling cylinder on a long straight ribbon, namely a space 

 periodicity with respect to z. But, if a particular relation subsists between 

 the parameters, ~ is a periodic function of lu, and the boundary is a triangle. 



Though the right-hand side of formula (22) is not a proper curve-factor, 

 the method of article 4 for obtaining the condition that z be periodic is 

 applicable to it. The condition is therefore the vanishing of the absolute 

 term in that expansion of dz/dw in ascending powers of exp {'2niw/\) which 

 is valid for great positive values of -v^. On putting 



■ '^ , \ I ■ ( '^'^ / \ I ( 1 ^ITT , ) 



sin - [10 - a) = ^i exp j — (o - w) [ si - exp ^— (w - «) [ , 

 A ( A ) ( A ) 



and employing the binomial theorem, it is seen that the expansion in 

 question is 



-l-iTexp |-2a^l --jj[exp^-^j-2(l " "j -p(-^j+ • • -J, 



(23) 



where the terms that should follow the final plus sign are positive powers of 



exp (2iriwlX). 



[7M 



