INFINITESIMAL PROPERTIES OF LINES IN S\ WITH 

 APPLICATIONS TO CIRCLES IN S^. 



By C. L. E. Moore. 

 Presented by H. W. Tyler. November 9, 1910. Received November 12, 1910. 



Introduction. 



In a recent paper (to be published in the American Journal, 1911) 

 the author discussed properties of systems of lines in *S4 involving dif- 

 ferentials of the first order, that is, simple tangency, and interpreted the 

 results in the geometry of the circle in space of three dimensions. It is 

 the purpose of the present paper to discuss properties of lines and 

 circles involving second and higher differentials. The methods of the 

 first paper are not applicable to this case. Here lines are represented 

 by points of a six-dimensional spread which stands in a space of nine 

 dimensions. The problem is, then, simply one of differential geometry in 

 higher space. 



1. The coordinates.^ In S^ let the line be defined by two points 

 (homogeneous coordinates). Then the coordinates of a line are the 

 two row determinants of the matrix, 



^l X2 X^ OC^ Xk^ 



yi 2/2 yz 2/4 2/5 



Adopting the usual notation 



Pik = a^iVk — ocuyu 

 we see that there are ten coordinates, since 



Pik = — Pkh Pa = 0. 



These ten coordinates are connected by the five quadratic relations 



(0 ^i = PnPmn + PkmPnl + PknPlm = 



^ Castelnuovo, "Ricerche di geometria della retta nello spazio a quattro 

 dimensione," Atti del Reale Istituto Veneto, series 7, 2, 1890-91. In this 

 paper only linear systems of lines have been considered. 



