196 Mr TurnbuU, Some Geometrical Interpretations 



Some Geometrical Interpretations of the Concomitants of Tivo 

 Quadrics. By H. W. Turnbull, M.A. 



(Communicated by Mr G. H. Hardy.) 



[Received 6 July 1918; read 28 October 1918.] 



§ 1. In the Mathematische Annaleii, Vol. LVi, Gordan has given 

 a system of 580 invariants for two quaternary quadratics. It appears 

 that by carrying out the processes of reduction a little further, the 

 irreducible forms can be shewn to number 123 at most. That is to 

 say, the system is about as complicated as the ternary system for 

 three conies which Ciamberlini* first established. It is therefore 

 worth while to give geometrical interpretations to members of the 

 system for two quadratics. In the following pages about a hundred 

 of them are shewn. The geometrical significance of the residue 

 appears to be remote. 



Using the classification introduced by Gordan, the numbers of 

 forms of each type J which have not been reduced are shewn in 

 the subjoined Table. The rows of the Table give the numbers of 

 forms of each particular order in the three sets of coordinates x, 

 p, u, which define points, straight lines, and planes respectively. 

 Detailed lists of these forms will be found at the heads of the 

 paragraphs which deal with separate types. 



Ciamberlini, Giornale di Matematiche, Vol. 



