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XIV. Some Applications of C (informal Transformation to Problems in 



Hydrodynamics. 



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

 Communicated by Sir JOSEPH L ARMOR, M.P., F.R.S. 



Received May 13, 1915. 



CONTENTS. 



Page 



1-.'!. Introduction 439 



4-12. Conformal curve-factors 441 



13-20. Hydrodynamical illustrations 447 



21-22. Inflexional curve-factors 456 



23-26. Curve-factors of semi-infinite linear range 458 



27-29. Double curve-factors 461 



30-33. Synthesis of curve-factors from elementary types 463 



34-41. Liquid motions with free stream-lines 467 



42-43. General remarks on curve-factors 475 



44-48. Curve-factors regarded as the limits of products of Schwarzian factors 477 



49-51. Transformations involving both variables explicitly 481 



52-55. Supplementary note on curve-factors 485 



INTRODUCTION. 



1. THE problem of the confonnal representation of the part of the plane of a variable z, 

 which is bounded by a rectilineal polygon, upon the half-plane of a variable w bounded 

 by the real axis, is solved (save for an integration) by the well-known transformation 



of SCHWARZ 



dz = CU(w~a r )-"' l "dw, 







where C, 1; a 2 , &c., are real constants, and TT OL^ TT 2 , &c., are the internal angles 

 of the rectilineal polygon. 



A more difficult problem is that of the confonnal representation upon the half-plane 

 of w of a region in the z plane whose boundary is partly curved ; it is with this 

 problem that the present paper is concerned, always however with a view to inter- 

 pretation of results in terms of the two-dimensional flow of liquid in regions having 

 particular types of boundary. 



VOL. COX V. A 536. 3 N [Published September 22, 1916. 



