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IV. 



ON PEEIODTC CONFORMAL CURVE-FACTOES AND 

 CORNER-FACTORS. 



By J. G. LEATHEM, M.A., D.Sc. 



Read February 14. Published August 11, 1916. 



1. Introduction. — In a previous paper* the writer has defined conformal 

 curve-factors, and exemplified their use in the conformal representation of 

 simply connected two-dimensional regions of assigned type, say in the plane 

 of a complex variable z^ x + iy, upon the principal half -plane of a variable 

 w = <!> + iip-. 



If there is justification for the hope that the method of curve-factors 

 constitutes a more systematic and comprehensive mode of approach to those 

 classes of physical problems which can be formulated in terms of conformal 

 transformation than any previously recognised method, it is worth while to 

 consider how it may be extended to the conformal representation upon the 

 principal half-plane of vj of such a doubly connected region in the z plane as 

 is unbounded externally but is bounded internally by a single closed curve, 

 not necessarily free from corners. Such a representation would find illustra- 

 tion in the circulatory irrotational motion of liquid round a fixed internal 

 boundary, the velocity being the downward gradient of tp, or in the electro- 

 static field round a charged cylindrical conductor, the electrostatic potential 

 being - yjr. 



If the hydrodynamical circulation round the cylinder, or alternatively its 

 electric charge per unit length, is to be definite, the inner boundary of the 

 field of flow or induction will correspond, not to the whole real axis in the 

 plane of u', but to a definite length X upon it, which may be called the " linear 

 period." The complete half-plane of tu on the positive side of the real axis 

 corresponds to the doubly connected region in the z plane, repeated again and 

 again, and s is a periodic function of iv having the real wave-length or linear 

 period X. Also dzldtv is a periodic function of w. 



The transformation, in its differential form, is therefore of the type 



* Some Applications of conformal transformation to problems in Hydrodynamics. 

 Roy. Soc. Phil. Trans. A., vol. ccxv, 1915, p. 439. 



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