( 499 ) 



quadratic cones degenerated into pairs of planes with respect to the 

 two sextriples of pairs of lines {ai,a'i), {b,b') and {cii, ai),{c,c') and. 

 therefore Pj lies on 0^^ and OJ*^ without being vertex of a quadratic 



cone with the transversals to the seven pairs («i,a'j), {h,b'), {c,c') 

 as edges. So each of the quadruples out of the five pairs of lines 

 {ai,ai) furnishes a o'^ common to O,*^ and O*^ but not corresponding 



to solutions of the problem; so the curve ^^^ found above consists 

 of these five curves o!^ which are to be discarded and the locus 



proper q^*\ 



The result (j^^' is easily checked as follows. Starting from seven 

 pairs of intersecting lines for which Ai and «,, (i = l,2, ..,7) 

 represent the seven points of intersection and connecting planes, the 

 locus consists of: 



1. the locus Q^ of the vertices of the cones contained in the net 

 of surfaces 0'^ through the seven points Ai, 



2. the section c" of aj\y of the seven planes ni with the surface 

 0* forming the locus of the cones through the six points A with a 

 subscript different from z, counted twice, 



3. the lines of intersection of the seven planes «,• by two, counted 

 four times. 



So we find 6 + 7. 4. 2 + 21. 1.4 = 146. 



The necessity of discarding a part of the completing intersection, 

 on account of tlie existence of a locus of points P for which the 

 cones C'l and C" corresponding to the systems {<ti,(ii), (h, //) and 

 {ni,a'{), (c,r/) break up into a common part C'' and two different 

 completing parts Cl '' and C'!~'' , presents itself in the case ?i z= 2 

 only. ¥ov this locus puts in its appearance under the condition 



[èMP + 3) + 2j -\- \{n-p){n-p + ^6)^l\n{n^^^\] 

 only ; for then the locus of the point P for which hp' p-\-2t) -j- 2 



transversals lie on a cone C'' furnishes a curve common to (7^ and 0\^ 

 the points of which do not satisfy the conditions of the problem. 

 As this equation reduces itself to p{n — p)-=l the only possible case 

 is p=l, n=i2. 



We now pass to a consideration of the cases 7i ]> 2 and take 

 ?z = 3 as example. Here the two surfaces {P)b and {P)c corresponding 

 to the systems K, a',), (6,/^') and [ai, a!i),{c,c'), where i goes from 

 one to nine included, admit besides the 18 common threefold lines 

 and the 72 common single lines a common twisted curve not con- 

 nected v\ith solutions of the problem, i.e. the curve forming with 

 the two groups of 18 and 72 lines the locus of the point P emitting 



