( ''i3 ) 



(leliiiitc wav, and IVuni four arliitrary planes a^, «,, <?,, «^. If ilien the 

 peneils uf ravs in (Ikpsc jdanes lying perspecüvely to the qnadratic 

 involutions of points of the sections are taken as the projectively 

 related pencils of rays of I he problem, then the surface O" is evidently 

 the integrating part of the corres[)onding surface O^, so this must 

 really also be completeil to a surface of order eight by three other 

 reguli (>^ To conform this we allow an analytical treatment to 

 follow this geometrical consideration. 



We suppose the locus pro[)er (P to be decomposeil into its genera- 

 tors by means of ti)e equations 



r + ;..-' = i 



and we assume that the generators l)elonging to ). = and to A = oo 

 represent the double rays of the tpiadratic involution on (P, i.e. that 

 in this involution the rays with the same absolute value of X correspond 

 to each other. Here p, q, r, s are general linear forms in x, y, z, 

 according to the formula 



n = u^x 4- u^y -\- »,5 + u^ , {u z= p, q, r, .s), 



whilst the three planes of coordinates .1: = 0, y = 0, j ::= and the plane 

 at inlinity will do duty for the planes a^, «„, «,, «^ of the pencils of 

 rays to be found. The tracing of these pencils is simplified by repre- 

 senting the minors of the determinant 



A=: 



/': 



Pa 



according to the elements p; , q, , /■, , s, by Pi , Qi , /?/ , ,S'/. 



If we perform the described calculation with respect to the plane 

 d- = {), there is occasion to represent the equations 



(}\ + ^--h) II + (y. + ^'7») - + lU + ^'7, = <J 



(»'. + ^*J // + ('•» + ^^,) = + >;+ ).,^ = 



determining together the point belonging to A of the conic of the 

 section, for shortness'sake by 



(p + kq), = I 



Then 



