Leathem — Periodic Conformal Curve-Faciors and Corner-Factors. 39 



J (Jdw round any contour in that region is zero. Let the contour be a rectangle 

 whose corners are 



^0 + 'i'/'i. ^0 + A + ti/zi, 0„ + A + i-^u (j)„ + iip, ; 

 through its periodicity the subject of integration lias equal values at corre- 

 sponding points on the sides lying in the lines ^ = ^u, ^ = ^o + A, and so the 

 corresponding contributions to the contour integral vanish ; hence the parts 

 of the integral corresponding to the sides in the lines i/^ = t/-,, <^ = \p2, add up 

 to zero. Thus / ^chv has the same value when taken from -^ (p^ to ^ = </>„ + A 

 with constant i^, whether \p = \pi or i// = »//■>. It follows that the mean value of ff 

 is the same for both ranges. 



This suggests a method of formulating the condition for a closed curve, 

 that is the condition for periodicity of s or J (odtv, which is useful in many 

 cases. It consists in getting the mean value of ^ for i// -> 4 od , and equating 

 it to zero. Suppose that, for great positive values of \p, G can be expressed as 

 a series of descending integral powers of exp (- 27rMy/A), say 



G = exp (- 2Trniw/\) "^ Cj exp {2'7rsvw/\), 



(11) 



and that the series is integrablo for tp ->- + co . Then, n being an integer, it 

 is to be observed that every term is periodic and has the mean value zero 

 except that corresponding to s = 7i, which is a constant. Thus the mean value 

 of 6^ is c„, and the condition for periodicity of j Gdw is c„ = 0. 



By way of illustration, let the test be applied to G^!"cn- This can be put 

 in the form 



kh'" exp (- 2Triw/\) [1 + A-' exp ( 27r (n - 1) iw/\ ] ] '/", (12) 



and the binomial expansion is valid for great positive values of i//. If n = 2, 

 there is a constant term in the expansion, but for any other integral value of n 

 there is none. Thus the transformation dz = G^'i"mdw gives a closed curve 

 for any integral value of n exeept ?i = 2. 

 Similarly 



^'/". = (i)''" exp 



2Triw a 

 A n 



1 -exp 



4:Ttniw 



(13) 



and the binomial expansion is valid for great positive values of ^. There 

 is no constant term for any integral value of n, and so the transformation 

 dz = G^l"is dw gives a closed curve. 



5. Relation hetween angular period and exponential order at infinity. — 

 When a periodic curve-factor can be expanded, for i// great and positive, in 

 the' form indicated in formula 11, it may be said to have a definite "expo- 

 nential order at infinity," namely 2mrlX, this being the coefficient of i/, in the 



