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The double points of such an involution are harmonically separated 

 by two surfaces chosen arbitrarily- out of {Q^) and two surfaces 

 taken arbitrarily out of {Q^)' . The locus of those double points is 

 therefore the surface of Jacobi belonging to those four surfaces. If 

 a'^j. = 0, l/x = t), c% = 0, d'^j- = are the equations of the indicated 

 surfaces, then for a pair of double points X, Y we have 



SO that the surface of Jacobi is represented by 



«i«x ^^x c^Cx d^dx I 



0, 



(1) 



a^a 



a^-x 



a^ax 



''i^X 



h,hx 

 h,hx 





d^dj,- 

 d^dx 

 d^dx 



= 



tjUj; 



h,hx 



Khx 



= 



(2) 



To find the number of pairs A', 1^ lying in a plane we put in (1) 

 x^ = 0, 11^ = 0. By elimination of yi, ij^, ?/., we then find the conditions 



a^üj- b,br c^Cx d^dx 



«3«x b^hj: c^Cx d^dx 



The determinants arising from this matrix, if one omits the third 

 or the fourth column, disappear for the points of intersection of two 

 twisted curves; to these belong the three points, for wiiich the matrix 

 of the first two columns disappears. The four twisted curves indicated 

 by \2) have thus six points in common forming three j)airs of X Y. 

 In an arbitrary plane Wo, {\\QYeïovQ three singular bmcants oï\\\q sqq.o\\& 

 species. 



From this ensues that the quadruple involution in which F is 

 cut by any plane contains fifteen singular lines; this corresponds to 

 a result obtained by me in another research ^). 



4. We consider the bisecants sent out by the curves (/ through 

 a given point P and w^e determine the surface 77 on which their 

 points of intersection lie. The q^ passing through P is projected out 

 of P by a cubic cone of which the generatrices touch /Z in P; so 

 P is a three fold point (triconic point) of n. An arbitrary line through 

 P is bisecant of one <)'* ; so /7 is a surface of order five with tri- 

 conic point P. 



The cone of contact for P can have with 77% besides the q' 

 through P, only straight lines in common. Hence through P pass 

 eleven singular bisecants. 



1) See my paper: 'A quadruple involution in the plane and a triple involution 

 connected with it" (Proc. of Arast. Vol. Ill p. 84). 



