80 Journal of the Mitchell Society. [June 



of a cubic surface. Indeed, it is feasible to derive the proper- 

 ties of one configuration from the known properties of the 

 other.* 



In 1877 Crenionat first showed that the Pascalian configu- 

 ration might be derived from the configuration of the twenty- 

 one lines upon the surface of the third degree with one coni- 

 cal point (Species II., Cayley's enumeration) by projection 

 from the conical point. 



The theory of varieties of the third order, that is to say, 

 curved geometric forms of three dimensions contained in a 

 space of four dimensions, has been the subject of a profound 

 memoir by Corrado Segre. t The depth of this paper is 

 evinced by the fact that a large proportion of the propositions 

 upon the plane quartic and its bitangents, Pascal's theorem, 

 the cubic surface and its twenty-seven straight lines, Rum- 

 mer's surface and its configuration of sixteen singular points 

 and planes, and on the connection between these figures are 

 derivable from propositions relating to Segre's cubic variety, 

 and the figure of six points or spaces from which it springs.f 

 Other investigators on this beautiful and important locus in 

 space of four dimensions and some of its consequences are 

 Castelnuovo and Richmond. § 



The problem of the twenty-seven lines is full of interest 

 from the group theoretic standpoint. In 1869Camille Jordan|| 

 first proved that the group of the problem of the trisection of 

 hyperelliptic functions of the first order is isomorphic with 



*Crelle's Journal, Vol. 132 (1900), pp. 209-226. 



tReale Accademia dei Lincei, Anno CCLXXIV. (1876-77). Roma. Also 

 cf. infra, §§47, 48. 



lTAtti d. R. Accad. di Scienze di Torina, Vol. XXII. (1887), pp. 547-557. 

 Memorie d. R. Accad. di Scienze di Torino, Series 2, Vol. XXXIX. 

 (1889), pp. 3-48. 



tRichruond, Quarterly Journal, Vol. XXXIV. No. 2 (1902), pp. 117-154. 



§Of . Richmond 1. c. for references. 



HComptes Rendus, 1869. Of. also Traite des Substitutions, p. 216 et seq., 

 p. 365 et seq. 



