( 498 ) 



3. Have we been able to deuuce until now theorems holding for 

 an arbitrary value of n, in proceeding to the determination of the 

 twisted curve q forming the locus of the point P, emitting transversals 



lying on a cone C' to — ?i (?i-|-3) -|- 2 pairs of lines [ni^a'i), {h,h'), 



{c,c') we are obliged to treat the cases 7z = 2, ?zrr: 3, etc. separately. 

 We will indicate first what is the cause of this and restrict ourselves 

 then in this communication to the case ?z = 2. 



The surfaces {P)b and (P)c-, corresponding in the manner indicated 

 in theorem I to the systems {ai,a'i), [b,b') and {a;, a-,), (c, c'), admit 



as such the — ?2 (?2-}-3) pairs of lines oj^a'/as common lines of 



1 

 multiplicity n and the — - n {n-\-'i) 



' 1 



transversals 



2 



cutting these pairs by two as common single lines. So these 

 surfaces intersect each other still in a curve of order 



4 1 



— n' (.n4- 1 y (n + 2)^ — n' (n | 3) n {n \ 3) {n- + 3n - 2) 



9 4 



— — n (n-f-l) (16n' + 80n'^ 83«' — 53n + 54). 

 36 



If now we had the certainty that each point P of this completing 

 intersection was the vertex of a cone C" with the transversals emitted 



by this point to the —- n {n-\-^) -\- 2 pairs of given lines as edges, 



the number indicated just now would represent the order of the 

 curve Q under discussion. This however is only the case for nr=l 

 where tiie obtained result passes into a o^°, as it ought to do (see 

 the paper quoted). For in the case of higher values of n the com- 

 pleting intersection found above consists of two or more parts, one 

 or more of which do not belong to the locus. In order to show this 

 we must treat the two cases ?2 ^ 2 and 7z ]> 2 separately. 



For n=z2 the tw^o surfaces 0!^ and 0'^^ have still in common 



c 



besides the ten common double lines and the twenty common single 

 lines the five twisted curves 9^° — as we shall see immediately not 

 connected with solutions of the problem — which form the loci of 

 the point emitting complanar transversals to four of the five pairs 

 {ai,a'i). Let F^ be a point emitting to the four pairs {<i^,a',), {a^,a\), 

 (a^, a\), (ötj, a\) four transversals lying in the plane «^ and let /?, and 

 Yi represent the planes of the pairs of transversals from P, to 

 (a,, a,'), {f),h') and (a,, a\), {ex); then {a„ ^,) and (rr^, y^) represent 



