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XXVIII. On Two-Dimensional Fields of Flow, with Loga- 

 rithmic Singularities and Free Boundaries. By J. G. 

 Leathem, M.A., D.Sc* 



1. JNTRODUCTION.— In a recently published paper f 

 the writer has defined conformal curve-factors, 

 discussed some of their properties, and employed them in 

 the solution of problems of two-dimensional liquid flow. In 

 all the configurations there discussed the fixed boundary has 

 been an open polygon, rectilineal or partly curvilinear, 

 whose continuity is not broken at any point, and the field 

 has been free from singularities. It is now proposed to 

 show how the same calculus may be applied to regions 

 whose fixed boundary is broken and to fields which contain 

 logarithmic singularities, as, for example, to cases of liquid 

 flow between two boundary stream-lines and cases of flow 

 due to sources or vortices. 



Methods of solving problems of this kind, with a special 

 view to the determination of the forms of free stream-lines, 

 have been given by Mr. J. W. Michell J, Prof. A. E. H. 

 Love §, and Prof. B. Hopkinson ||, who all deal with cases in 

 which the fixed parts of the boundary are rectilineal polygons. 

 It will be shown that the method of curve-factors deals with 

 such problems in a different and more concise manner, and 

 is further applicable to some cases in which part of the fixed 

 boundary is curvilinear. The analysis is, of course, capable 

 of interpretation in terms of fields of electric flow or force, 

 and it is believed that the method of curve-factors may be 

 regarded as a comprehensive mode of approaching all those 

 classes of physical problems which can be formulated in 

 terms of conformal transformation. 



2. Double transformation. — The general problem is that of 

 determining a relation between two fundamental complex 

 variables, namely, z = x + iy the variable of the geometrical 

 configuration, and w = cf> + iyjr the variable specifying the 

 field of flow or force (the velocity being the vectorial rate of 

 decrease of </>). When the field is free from singularities 



* Communicated by the Author. 



t " Applications of Conformal Transformation to Problems in Hydro- 

 dynamics," P. S. Phil. Trans., sec. A, vol. ccxv. (1915). 



X " The Theory of Free Stream-Lines," R. S. Phil. Trans., sec. A, 

 vol. clxxxi. (1890). 



§ " The Theory of Discontinuous Fluid Motions," Proc. Camb. Phil. 

 Soc. vol. vii. (1891). 



|| " Discontinuous Fluid Motions involving Sources and Vortices," 

 Proc. Lond. Math. Soc. vol. xxix. (1898). 



