This volume, is concerned with a locus - itself very interesting to explore geometrically - which exhibits in a simple way the structure of the group of the lines of a cubic surface in ordinary space, regarded as the group of the tritangent planes of the surface. Incidentally certain quite elementary results for the substitutions of 4, 5 and 6 objects are necessary; and, for the sake of completeness, these are explained in detail. Historically the linear expression for the group of transformations considered, and of the different linear expression briefly sketched in Note 2, of the Appendix, both arose from the theory of the linear transformations of the periods of theta functions of two variables; but, beyond references to the literature, this is not dealt with. It is hoped that the Introduction to the text, and the list of headings of the sections, will make sufficiently clear what is included. A brief index of notations is also appended. The argument of the text requires frequent reference to the Scheme of synthemes given as frontispiece of the volume.
To some it may seem that such a theme - at this time - is futile. It is possible, however, to take the view that the primary purpose of the pursuit of science is not the advancement of technology, but the widening of the horizon of the human mind. In this mathematics has always borne an honourable, often a decisive part; indeed, many cases could be cited to support the more extreme view that the development of mathematical ideas and the emergence of new physical conceptions, are intimately related.
The general theory of linear groups has developed widely on the algebraic side since the group considered here - one of the earliest - was established; and even of this the present is a very incomplete account. The writer has had the advantage of the co-operation, in the reading of the proof-sheets, of Dr J A Todd, who has, himself written on the matter, and is very greatly indebted to him. He would wish also to acknowledge his obligation to the Staff of the University Press, especially at this time of difficulty in the printing of books.
H. F. B.
26 November 1945
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