36 Proceedings of the Royal Irkh Academy. 



dz = (^{xi))dvj, where G{v:>) is a curve-factor periodic in ^ with linear period A. 

 And, as a tangent to the internal boundaiy makes one complete revolution as 

 the point of contact goes once round the boundary, the angular sub-range of 

 G corresponding to the linear sub-range A is 27r ; this fact may be conveniently 

 expressed by saying that the " angular period " of (^ is lir. 



It is proposed to look for types of periodic curve-factors which can be 

 employed to give conformal representation of doubly connected regions whose 

 only boundary is internal, and it will be seen that such curve-factors may be 

 used to construct not only differential transformation formulae but also 

 formulae in which z is expressed explicitly in terms of w. 



2. Circular and elliptic curve-factors and derived types. — "\Anien the Ijoundary 

 is a circle of radius a, the origin of z may be taken at the centre, and the field 

 of flow or induction is determined by the relation 



W = (iX/27r)l0g(2/«), (1) 



where the sign is so chosen that <f> increases as the circle is described in the 

 clockwise sense, which implies keeping the relevant region on the left. 

 The relation is ee^uivalent to the differential formula 



(h = - (2-n-m/A) exp (- 2iriwlX)dw, (2) 



and this gives the periodic curve-factor 



^57 = exp (- 2iriwl\). (3) 



When the boundary is an ellipse of semi-axes ccosh a, csinh a, with centre 

 at the origin of z and major axis along the real axis, the field is determined by 

 the relation 



z = ccosh [- {2irij\) w + a]. (4) 



The corresponding differential relation, 



dz = - (27ric/X) sinh [- (2-iri/X) w + a], (5) 



gives the curve-factor 



G^ = sinh [- (27rt/A) w + a']. (6) 



(tbt and (f-.i have no zeroes or infinities for definite positive values of i^. 

 Their only infinity in the relevant region is for -\/r-> -t- oo , and that of course 

 corresponds to the external boundlessness of the relevant region in the z plane. 



As periodicity with a linear period which is a submultiple of A implies 

 periodicity with linear period A, the substitution in d-, and (fi^ of Xjn for A, 

 where n is any integer, will give periodic curve-factors. In the case of 6^57 the 

 substitution leads simply to the nth. power of /fi-,, and so does not give a new 

 type. But 



(fi, = sinh [- (27r«i; A) %u + a] ' (7) 



is not a mere power of Gk and is therefore a new type. Its angular period 



