Lea THEM — Periodic Conformal Curve-Factors and Corner-Factors. 43 



value ; hence, as each contributes a factor exp (- ina), it is necessary that 



exp |- iw («„ + 22a)| = 1, or Oo + 2Sa = an even integer. 



In this statement, on account of the exceptional circumstances at 0, an 

 integral odd power of H must be regarded as a branching; thus, to take the 

 simplest possible example, the function 



changes sign with 0, but the function 



does not. 



It is of interest to see how some of the already known types fit into the 

 formula. It is easy to see that 



e,-, ^ exp (- 2^nvl\) = 1 - 2 ^ - 2- f ^ - 1^, (15) 



and that 



^. -r--ic'- + 0(9=-r)4 (16) 



is a curve-factor in 6 of linear range - c to c and angular range 'Itt. Similarly 

 it is seen that 



ffs^s sinh (- (2'7riw/X) + a \ 



= f 1 - 2 ^} sinh a - 2 -j^- - ij cosh a, (17) 



where 



^03 - (O' - ic2) sinh a + e (0- - c2)2 cosh « (18) 



is a curve-factor in of linear range - c to c and angular range 27r. 

 Another suitable curve-factor in is 



Ai ^B(B'- ¥) V ((92 - ff2)4 (0^ - c-f, (19) 



where a<b<c; this is a special case of ^47. 



A special case of (''d, which has the advantage of being a simple curve- 

 factor, is 



tn,, ^£{d'- I') + 0(0'- c')^ (20) 



where h<c. This leads to the periodic curve-factor 



^60 - £ {cos (i-mvIX) - cos 7 j + i sin {2ttwj\). (21 ) 



The formulation of the present article suggests the question whether there 

 has been left open any possibility of a corner in the curve in the z diagram 

 which is defined by i// = 0, at the point (or points, if z is not periodic) corre- 

 sponding to 9 = + c. The answer is that if the curve-factor in is a proper 

 curve-factor there is no such corner. A curve-factor (/{()) may be described 

 as 'proper' when the transformation dz = (/{djdO gives in the z plane, as 

 corresponding to real, a locus without corners, that is a finite curved line 



i;.I.A. PEOC, VOL. XXXIII, SECT. A. [7] 



