IjKATHiiM — Periodic Conformal Curve- Paciors and Corner-Factors. 47 



This is as free from zeros and infinities in the relevant region as is Cti, and is 

 clearly periodic of linear period X. It is therefore a periodic curve-factor, and 

 has the angular period 2it. 



A curved boundary with one or more corners is given by the special form 

 assumed by (Tps when one or more of the constants a, /8', . . . is taken to be 

 zero. 



10. Periodic ctirve-fadors regarded as limits of 2^1'oducts of periodic corner- 

 factors. — The field outside a closed polygonal boundary being obtained by the 

 method of article 8, it is possible to increase the number of sides of the 

 polygon without limit in such manner that the polygon tends to a limit form 

 which is a smooth closed curve. The corresponding limit of the product of 

 corner-factors which takes the place of the right-hand side of formula (22) is 

 then a periodic curve-factor, and serves to define the field of flow or induction 

 outside the boundary curve. 



Attention being directed to such a smooth curved boundary, the angle 

 (measured in the clockwise sense) which a tangent to the curve makes with a 

 fixed direction may be called ^, and f^x/''' takes the place of 1 - (A/tt) as 

 index to the periodic corner-factor 



sin JTT (I'J - a)/X}. 



Here a is a real variable which is to be regarded as varying continuously 

 round the curve, increasing by A with each complete description of the curve 

 in the clockwise sense. The transformation then takes the form dz = Kffdw, 

 where 



C^limU (sin ~ (iv - a) j "^^'"^ exp ^ log sin j T (^^ " ") ! . (30) 



and, corresponding to formula (25), 



J cos (27r../A) dx = 0, / sin (27ra/A) dx = 0, (31) 



the integrals being taken over a range A of the variable a. ^ is a periodic 

 curve-factor. 



These formulae are indefinite until a functional relation is known or 

 assumed between a and x> say x ^/(")- With such a relation postulated, 

 and with the range of values of a specified as being from a = a to h = « + A, the 

 formulae take the definite shape : — 



a + \ 



^69 = exp - log sin ] T ('^ - «) f (") '^a. (32) 



a 

 '^"^^OS (27ra/A)/'(a) da = 0, p\in (27ra/A)/'(a) da = 0, (33j 



