246 REPORT— 1897. 



The Historical Developme^it of Ahelian Functions up to tJie time of 

 Riemann. By Harris Hancock, 



[Ordered by the General Committee to be printed in externa among the Reports] 



(1) In 1846, R. Leslie Ellis ' presented to the British Association a 

 * Report on the Recent Progress of Analysis (Theory of the Comparison of 

 Transcendents).' At the beginning of this memoir he says : 'The province 

 of analysis, to which the theory of elliptic functions belongs, has within the 

 last twenty years assumed a new aspect ; in no subject, I think, has our 

 knowledge advanced so far beyond the limits to which it was not long 

 since confined.' ' This circumstance,' he continues, ' would give a particular 

 interest to a history of the recent progress of the subject, even did it now 

 appear to have reached its full development. But on the contrary, there 

 is now more hope of further progress than at the commencement of the 

 period of which I have been speaking.' 



These statements appear more emphatic when we consider that after 

 the lapse of fifty years, since the publication of Ellis's report to the present 

 time, the same remarks are literally true, and when at the end of this 

 period we find that there is more hope for the future progress of analysis, 

 the theory of functions, than there has ever been before. 



So great has been the growth of this science, extending on the one hand, 

 and with a broadening influence, far into the realms of almost every 

 branch of mathematical study, and on the other hand so comprehensive 

 and varied in character is its application to physical problems, that the 

 development of Ellis's work must be divided into many parts. 



(2) The present report which the author has the honour of submitting 

 to the Association is intended as a brief account of that part of the work 

 already begun by Ellis which treats of the developments of the Abelian 

 (including the hyperelliptic) functions. It is also found that the develop- 

 ment of these functions has been so rapid and so extended that an ade- 

 quate account of it would require much more space than can be given here. 

 The author has consequently decided to make this statement for the period 

 up to the time of Riemann. With Riemann, Weierstrass, Clebsch and 

 Gordan, Cayley and others, the subjecttakesdirections so essentially different 

 that separate accounts along these different lines seem very desirable. 



Much regarding the history of the general theory of functions may be 

 found in Forsyth, ' Theory of Functions ' ; Harkness and Morley, ' A 

 Treatise of the Theory of Functions ' ; Casorati, ' Teorica delle funzioni di 

 variabili complesse ' ; Brill and Nother, ' Die Entwicklung der algebrai- 

 schen Functionen in alterer und neuerer Zeit ' (see ' Jahresbericht der 

 deutschen Mathematiker-Vereinigung,' 1894, bd. iii.). Fruitful sources 

 for researches regarding the elliptic functions are Konigsberger, ' Zur 

 Geschichte der Theorie der elliptischen Transcendenten in den Jahren 

 1826-29,' Leipzig, 1879 ; short notices about ih^ first discovery of elliptic 

 functions are given by Gauss, ' Werke,' iii. p. 491 ; ' Correspondance 

 math^matique entre Legendre et Jacobi ' (Crelle's Journal, bd. Ixxx. 

 p. 205) ; and especially good is the account given by Enneper, ' Elliptische 

 Functionen : Theorie und Geschichte,' Halle, 1890. 



These works give more or less extended accounts of the subject under 



' Ellis, Report of the Sriiish Association for the Advancement of Science, 1846, 

 p. 34. We shall hereafter use the word ' Ellis ' in referring to this paper. 



