( 491 ) 



trices Iviiiu' in «; moreover we sliall see directly tluit the number 

 of tliose nodes is in general six, becoming infinite when it surpasses 

 six, as takes place in the surface to be considered presently and 

 which is indicated in the title. 



5. We point out the fact, that the found surface 0" is determined 

 by the projective correspondence of the curves s^^ and .s'^^ in a^ and 

 «2, and we now show tliat this correspondence, characterised by the 

 particularity of the corresponding triplets lying on a^ andja^, is not 

 the most general one can think of. To that end we talve two rational 

 curves s^^ and ^2^ in two planes «i and a^, which planes for convenience' 

 sake we assume for the present to be lying in our space, and which 

 curves with the nodes 0^ and 0^ we suppose to be brought into 

 projective correspondence in the most general manner. Are there 

 then — we ask — to be found on 5^ Mh ree coliinear points to which 

 on s^^ three likewise coUinear points correspond? The answer runs 

 affirmatively; what is more: each point of .s'l" forms one time a 

 part of such a triplet and the bearing lines form a pencil of rays. 

 If namely the point A, of .s\^ corresponds to the point A^ taken 

 arbitrarily on s^\ and if the central involution of the points B^, C\ 

 of 5i' collinear with A^ is represented by (B^ C\), the non-central 

 involution of the corresponding points B^, C\ of s^^ by {B^ C\) and 

 the central involution of the points B.^, C^ of ó\/ collinear with A^ 

 by {B^ CJ), then the two involutions {B^ C.,), {B^ C\') have a pair 

 of points in common. If ^5^°, C° is this pair and B^°, C° on-y^Mhe 

 pair corresponding to it, then A,,B^°,C\° and A^,B^°,C^'' are two 

 corresponding collinear triplets. If now Q^ is the point of intersection 

 of two such like lines //, //' in «^ and Q^ the point of intersection 

 of the corresponding lines IJ, /," in a^, then the triple involution 

 {A^ B^ Cj) marked by the lines through Q^ in ^\^ must correspond 

 to the triple involution {A^ B^ C^) marked by the lines through 

 Q^ in .y/, with which we have proved what was asserted above. 



With the aid of the preceding it is easy to show in how far the 

 particularity of the corresponding triplets lying on a^ and a^ is a 

 real one or an apparent one. With respect to the planes «j and «^ 

 placed in our space it is evidently an apparent one ; for not one time 

 but an infinite number of times it happens that three collinear points 

 of >9i' correspond to three likewise collinear points of s^"^. If the 

 planes «^ and «„ are placed in Sp^ in such a way that an arbitrary 

 point I\ of «1 coincides with an arbitrary point P^ of a^, then 

 however the three points in which s^^ is cut by the line P^ Q will 

 correspond to three collinear points of s/, but the line through Q^ 



U ^ 



Proceedings Royal Acad. Amsterdam. Vol. VIII. 



