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Published online by Cambridge University Press: 20 January 2009
There is, in the second (Cambridge, 1911) edition of Burnside's Theory of Groups of Finite Order, an example on p. 371 which must have aroused the curiosity of many mathematicians; a quartic surface, invariant for a group of 24.5! collineations, appears without any indication of its provenance or any explanation of its remarkable property. The example teases, whether because Burnside, if he obtained the result from elsewhere, gives no reference, or because, if the result is original with him, it is difficult to conjecture the process by which he arrived at it. But the quartic form which, when equated to zero, gives the surface, appears, together with associated forms, in a paper by Maschke1, and it is fitting therefore to call both form and surface by his name.
page 93 note 1 Math. Annalen, 30 (1887), 496–515.CrossRefGoogle Scholar
page 94 note 1 Math. Annalen, 17 (1880), 510–516.CrossRefGoogle Scholar
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page 95 note 1 Klein, : Zur Theorie der Linienkoraplexe des ersten und zweiten Grades: Math. Annalen, 2 (1870), 198–226;CrossRefGoogle ScholarGesarnmelte Math. Abhandlungen, 1, 53–80.Google Scholar
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page 96 note 2 Elementary Researches in the Analysis of Combinatorial Aggregation : Phil. Mag., 24 (1844), 285–296;Google ScholarMathematical Papers, 1, 91–102.Google Scholar The 15 synthemes formed by Sylvester from a set of six objects are, when these objects are linear complexes mutually in involution, equivalent to Klein's 15 fundamental tetrahedra.
page 97 note 1 Über eine geometrische Repräsentation der Resolventen algebraischer Gleichungen : Math. A'nnalen, 4 (1871), 346–368;Google ScholarGesammelte Math. Abhandlungen, 2, 262–274.Google Scholar The fact that a sextic equation has a sextic resolvent distinct from the equation itself is due to the existence, first pointed out by Sylvester in 1844 in the paper to which reference has already been made above, of a function of six variables which, when the variables are permuted, takes six different values.
page 97 note 2 Kalkül der abzählenden Geometrie (Leipzig, 1879), 246.Google Scholar
page 99 note 1 The equations of all the faces and the coordinates of all the vertices of the fundamental tetrahedra, referred to one of them as tetrahedron of reference, are given by Hess, : loc. cit., 107–8.Google Scholar
page 100 note 1 Atti d. Reale Accad. di Scienze di Torino, 22 (1887), 791, where the dual configuration of 15 planes appears.Google Scholar
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