The aim of this translation is to reproduce, as far as possible, the ideas and style of the original in idiomatic English, rather than to give a literal rendering of its contents. Even the verbal deviations, however, are few in number. So little has been written in English on the subject that a standard set of technical terms as yet hardly exists. Where there was any choice between equivalent words, I have followed the usage of Dr Forsyth in his recently published work on the Theory of Functions. A Glossary of the principal technical terms is appended, giving the original German word together with the English adopted in the text. Prof Klein had originally intended to revise the proofs, but owing to his absence in America he kindly waived his right to do so, in order not to delay the publication. The proofs have therefore not been submitted to him, though it was with considerable reluctance that I determined to publish without this final revision.

My thanks are due to Miss C A Scott, D.Sc., Professor of Mathematics in Bryn Mawr College, for many valuable suggestions in difficult passages and for her interest in the progress of the translation, and also for help in the reading of the proof-sheets. I must also express my thanks to Mr James Harkness, M.A., Associate Professor of Mathematics in Bryn Mawr College, for helpful advice given from time to time; and to Miss P G Fawcett, of Newnham College, Cambridge, for reading over in manuscript the earlier parts which deal more especially with the subject of Electricity.

The origins of the theory of point groups are to be found in Brill and Noether's classic memoir published nearly twenty-five years ago, but it is only within the last fifteen years that systematic attention has been given to the subject by the Italian mathematicians, Segre, Bertini, Castelnuovo and others; in their hands a series of isolated problems bids fair to develop into an organic theory, a theory, moreover, which furnishes links in thought between subjects as apparently diverse as projective geometry and the analytical theory of elimination, while maintaining the initial, direct connection between the theory of higher plane curves and the theory of functions. In the following pages I have briefly indicated some of the converging lines of the German and Italian work; in the first section, by a discussion of certain of the technical terms; in the second section, starting from the so-called Riemann-Roch equations, by the suggestion of certain lines of enquiry which may prove useful in the classification of algebraic curves. The appended bibliography is divided into two parts; the first contains those memoirs which may be considered as fundamental, from the historical as well as from the purely technical point of view, to the theory of point groups, including those more especially concerned with linear systems of plane curves; the second includes memoirs on the theory of curves in ordinary and in hyperspace, more or less directly connected with the theory of point groups.

The term point-group is a direct translation of the German word 'Punktgruppe', first used by Brill and Noether in the year 1873 in their classic memoir on algebraic functions, but to my knowledge, although more than a quarter of a century has elapsed since then, there has been no very systematic attempt to present the theory of point-groups to English readers along any of its lines of development. And yet it should prove of interest even to those mathematicians who do not desire to specialise in it, for, historically and logically, it touches upon many distinct branches of pure mathematics. To mention only those which are most directly brought into connection with each other, we have the intersections of plane curves, the elimination of variables from systems of equations, the algebraic theory of correspondences on a plane curve, properties of linear systems of plane curves, and applications of the theory of functions to the theory of curves and surfaces in space of any number of dimensions.

As frequently happens when the progress of a subject has been due to many different writers, the logical and the chronological divisions do not coincide. I have therefore in view a dual arrangement of the subject-matter. In the present instalment of my Report, I have attempted to sketch this proposed arrangement under its two aspects, viz-; as an historical outline, and as an analysis according to content. This is followed by a detailed account of one of the historical, divisions. I hope in the subsequent portions of the Report to deal in a somewhat similar way with the remaining divisions, and to append a complete bibliography.

... it is clear that Frances interest in suffrage and women's rights reach back to her days at Girton where she spoke to the society on women's suffrage talking about the aims and history of and its relation to the Women's Suffrage Movement and the Higher Education of Women and was a member of the CWSA (Cambridge Association for Women's Suffrage). Later, Frances became Honorary Secretary of the National Union of Women's Suffrage Societies (NUWSS) under Millicent Fawcett and can be seen to sign a letter written in 1908 to 'The Times', stating her disagreement with militant methods taken by suffragettes.

1. To demand that international disputes shall in future be settled by some other means than war, and

2. To claim that women shall have a voice in the affairs of nations.

It seems that an accident, possibly in conjunction with ill health, caused Hardcastle to leave Cambridge in 1904 and abandon her mathematical research. She was able to live in adequate comfort on money settled on her by her father, increased later by inheritance. She and her sister Mira Francisca were named joint executors of her father's will, despite the availability of several brothers. She lived in London for a period, and worked for a year as a joint secretary, with Frances Sterling, for the National Union of Women's Suffrage Societies (the suffragist organisation), before moving to Newcastle where she served as secretary of the North-Eastern Federation of Women's Suffrage Societies. She spent her eventual retirement beside the Northumberland moors at Stocksfield, in a house she had built with Dr Ethel Williams, a former Newcastle GP.

... stating her disagreement with militant methods taken by suffragettes. Her role highlights many of the tensions within the movement head on and another example can be found in archive material showing her dealing with difficult correspondence with Scottish suffragists wishing to split and create their own division.