C " ] 



II. 



ON TWO-DIMENSIONAL FLUID MOTION, WITH FEEE STREAM- 

 LINES, PAST AN OBSTACLE OF CURVED OUTLINE. 



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



Received Auoi-st 23. Read Novembek 12, 1917. Published March 14, 191S. 



Introduction. 



1. There is already a considerable amount of literature on the subject of 

 the two-dimensional flow of infinite liquid, with free stream-lines, past a fixed 

 obstacle of curvilinear outline, or, what comes to the same thing, the motion 

 of such an obstacle through liquid which is "at rest at infinity." The 

 following papers are those which seem most relevant to the present 

 discussion ; they are arranged in chronological order, and reference will 

 be made to them by quoting the prefixed capital letters: — 



A. T. Levi-Civita. Scie e Lefjyi di Eesistcnza, Eendiconti del Circolo 

 Matematico di Palermo, xxiii, 1907. 



B. U. Cisotti. Veiie Fluenti, Eendiconti del Circolo Matematico di 

 Palermo, xxv, 1908. 



C. H. Villat. Sur la resistance des fluides, Annales de l'Ecole Normale 

 Superieure, 3me serie, xxviii, 1911. 



D. J. G. Leathem. Some applications of conformal transformation to 



problems in hydrodynamics, Phil. Trans. Royal Society, a, ccxiv, 1915. 



E. H. Levy. Discontinuous fluid motion past a curved boundary, Proc. 

 Eoyal Society, a, xcii, March, 1916. 



F. J. G. Leathem. On two-dimensional fields of flow, with logarithmic 

 singularities end free boundaries, Philosophical Magazine, xxxi. March, 1916. 



G. J. (J. Leathem. Theorems mi conformal transfomiation, Proc. Lond.« 

 Math. Soc. xvi, 1917. 



In these researches there have been three distinct objectives: — 

 (i) A mathematical formulation, in terms of somewhat general functions, 



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