﻿310 Sitzung der physikalisch-mathematischen Klasse 



T = (agh)x 2 -f- (bhf)y 2 -+- (cfg)z 2 + (abc)w 2 



+ [(<*bg) — (cah)]xw -f- [(bfg) + (chf)]yz 

 + [(6c/0 — (abf)]yw + [(cyA) + (afg)]zx 

 "+■ [(™/) — (&c$0]*w + [(aÄ/)+ (bgh)]xy = , 



where (agh) is used to denote the determinant 



«2 ,02 » Ä 2 , and 



so for the other Symbols. Considering the reciproeal of the line 

 L in regard to the quadric surface X 2 + Y 2 -+- Z 2 -4- W 2 = , the 

 six coordinates of the reciproeal line are 



f,g,h ,a ,b ,c , 



and it is hence at once seen that the locus of the reciproeal line 

 is the quadric surface obtained from the equation T = by in- 

 terchanging therein.the symbolical quantities a,b,c and f,g,h: 

 viz. writing also (£,*?,£,««) in place of (x ,y , z ,w), the new 

 equation is 



T f = (fbc)% + (gca)y? + (hab)? + (Jgh)»? 

 + VJgh) - Wc)]£* -f- [(/«*) + (höajijz 

 + [ty*?j - (fga)]iu 4- [fo&c) + (/a6)]^ 

 + [(hfa) - (ghb)]^ + [(hea) + (flfftc)]^ = ; 



or what is the same thing this equation T' = is the equation 

 of the original quadric surface (the locus of IS) expressed in terms 

 of the plane -coordinates f , v\ , £, w. 



Now considering each of the quantities fy , b y , c^ , f^ , g 1 , h x , 

 a 2 ,b 2 , etc., a 3 , b z , etc. as a given linear funetion of a variable para- 

 meter A, say ö^ = a{-+-öl'A , /^ = b[-hb['A , etc., the equation T= 

 takes the form 



A>? + 35A 2 + 3CX -h D = , 



where A , B , C , D are given quadric fuuetions of the coordinates 

 x,y,z,w; and the quadric surface T=0 is Herr Kummer's 

 surface of the eighth order 



(AD — BCf — 4 (AC — B 2 ) (BD — C 2 ) = o ; 



