Section III., 1913. [81] Trans. R.S.G. 



On Certain Difficulties that Arise in Connection with the Study 

 of Elliptic Functions. 



By J. Harkness, F.R.S.C. 



(Read May 28, 1913) 



In preparing material in connection with the article on Elliptic 

 Functions in the Encyklopadie der Mathematischen Wissenschaften 

 (EUiptische Funktionen mit Benutzung von Vorarbeiten und 

 Ausarbeitungen der Herren J. Harkness in Montreal, Canada, und 

 W. Wirtinger in Wien von R. Fricke in Braunschweig) the writer was 

 impressed by the inadequacy of the treatment of fundamental ques- 

 tions in many of the text-books, and by the scattered nature of the 

 material in cases where no objection is to be raised on the score of 

 logical rigour. Also methods of great importance for the particular 

 subject of elliptic functions are more or less masked by being pre- 

 sented in connection with the larger theories of multiply periodic 

 functions of many variables, or by being used incidentally in connec- 

 tion with problems of a different character. It has accordingly seemed 

 to him that it might be useful to discuss, from the most modern 

 point of view, some of the difficulties that arise in treating of the 

 inversion problem and the subject of systems of periods, and to show 

 how they have been overcome. It has not been thought necessary 

 to include many references to authorities, 



THE inversion PROBLEM. 



Let the basis equation be 



y2=a(x-ai) (x-aj) (x-ag) (x-aj, 



where the a's will be supposed finite and distinct, but capable of taking 

 complex values. The Riemann surface T attached to this equation 

 may be constructed in the ordinary way by connecting the two sheets 

 along two branch-cuts, going e,g, from ai to aj and from ag to a;^. On 

 this surface the pair of cross-cuts A, B, may be drawn in a variety of 

 ways. For instance, A may be taken on the upper sheet round the 

 branch-cut a^ ag and B round aj ag so that it lies partly in the upper 

 and partly in the lower sheet. The surface T, when dissected by A, B, 

 becomes a simply connected surface T\ bounded by a single contour. 

 In the second of the accompanying figures the two banks of A, B aie 



