Leathem — Periodic Conformal Curve- Factors and Corner^Factors. 37 



corresponding to the linear sub-range A is 2mT, so that if an angular period of 

 'Itt were desired (r^V'sg might be employed. 



One of the important characteristics of the special kind of conformal 

 representation now under consideration, namely the periodicity of dz/dw with 

 linear period A and angular period Stt, can be secured by making dz/dw pro- 

 portional to 6'^l'\g or to a product of powers of two or all of the types 

 (?57, 6^58, <ff59, provided the angular period of the combination is 2-n: For 

 example, ff^'si ^Xs or ^'''ss fo^^ is, to this extent, applicable provided p + nq = 1. 



The sum of two curve-factors is sometimes, but not always, a curve-factor. 

 For example, the function 



^60 = exp (- 27rMii/A) + h exp (- 2'?rniiv/\), (8) 



can only vanish when the moduli of the two complex terms are equal, so that 

 exp {2ir\pjX) =1^1 exp (2n7r->|r/A) ; for positive t/t this implies 



1 = I Z; I exp {2(n- l)7r-f/A|, 



which is impossible if | Z; | > 1. Hence, when this inequality is satisfied, 

 fff,,, is a curve-factor. As regards angular period, if one represents the two 

 terms as vectors which are to be added by the triangle law, one readily sees 

 that the angular period of the sum is the same as that of the term which has 

 always the greater modulus. So the angular period of C^ is 2mr. 

 Similarly it can be seen that 



Cn = a exp (- 2Trn,iw/X) + b exp (- 2'irniiw/\) + c exp (- 27r?i3W^) (9) 



is a curve-factor provided that, for all positive values of yjr, 



I a I exp {2-7r7ii-\jr/X) > \ i \ exp (27rmj-\/r/A ) -i- | c | exp (27rM3-\|r/A), (10) 



as, for example, when n^ > ih, ?i, > n,, and \ a \ > | & | + | c | ; the 

 angular period is 2?ii7r. 



3. Transformations whicli are not in differential form. — There is another 

 way of employing periodic curve-factors for obtaining conformal representa- 

 tion of regions of the kind under discussion. If such a representation be 

 specified by a formula z =f{w), the origin of z being supposed inside the 

 closed boundary, the function/ has to satisfy three requirements : — (1)/ must 

 have no zeroes or infinities for positive values of y^, save an infinity for 

 ■\|r->- + 00 . (2) /must be periodic in w with real linear period A. (3) The 

 periodicity of /must be such that, when a point traverses a length A of the 

 real axis in the w plane, the corresponding point in the z plane describes a 

 closed path which encircles the origin once and only once. Now any periodic 

 curve-factor G [w), whose linear and angular periods are A and 27r respec- 

 tively, satisfies all these requirements. Hence z = G{w) specifies a conformal 

 representation of the kind of region desired. 



[6*] 



