( 497 ) 



case of the projectively related pointfields ct^,a^ on the locus of 

 the line P^P^_ connecting corresponding points /*. , P^ of the planes 

 «J, «2, which is a quadratic or a cubic space according to the point 

 of intersection P^^ of «i and «^ corresponding to itself or not. 



13. We conclude with the deduction of the equations of the above 

 found cubic and quadratic spaces which have in common the 

 surface 0^ with the nodal curve k'^ and to this end we start from 

 this curve. If the curve k"^ is given by the system of equations: 



^.r;=;.sO- = 0, 1,2, 3, 4). (1) 



— and in this way the simplex of coordinates can always be taken — , 

 and if the point which is the vertex of the quadratic conic space 

 with respect to that same simplex has the coordinates (^o, ?/j, y^, ?/3, ?/ J, 

 then the equations 



il'n X, iV, 



/i\ tV„ ,^', 



= 



tï/'n tS. it' A', 



Vo Vi y-, Vz 



= 



(2) 



Vi y^ yz y, 



represent those t^vo spaces. We see namely immediately that the 

 first determinant by insertion of the relations (1' shows three equal 

 rows, i.e. that the cubic space represented by the tirst equation must 

 have the points of the curve k'^ as nodes, and must thus contain each 

 bisecant of k\ Further it is equally clear that the second deter- 

 minant by insertion of the relations (1) shows two equal rows and 

 that, when substituting yi for .17 , two pairs of equal rows appear, from 

 wiiich ensues that the quadratic space represented by the second 

 equation passes through k^ and has a node in ?/. 



A. more direct deduction of the equation of the locus of the 

 bisecants of the curve k'^ was communicated formerly {Proceedings 

 of the February meeting of 1899 vol. I, page 313). It is founded 

 on the wellknown lemma, according to which the product of two 

 matrices J/i''^' and J//'^' with r rows and k columns, taken according 

 to the rows, vanishes identically for r ^ k. This same lemma leads 

 to the deduction of the equation of the locus of the planes con- 

 taining three points of k\ and passing through (j/oy yi> J/^y l/s^ l/i)- ^^ 

 arbitrary point P of the plane P^ P^ P^ through the points P^, P^, P^ 

 of k"^ corresponding to the parametervalues ?.^, X^, ).^ is represented by 

 Q ^i = Pi ^-i^' + V. K' + Pz K' . (^' = 0, 1, 2, 3, 4) . . . (3) 



If the plane P^ P^ P^ passes moreover through the given point 

 U<^^lli^y^^y^^yA)^ ^"^^^ tlie relations 



