442 EEPORT— 1881. 



The other case (h < h^) is much more difficult. It is only wheu h haa 

 its minimum value that tlie problem can be easily solved, because the 

 curve lies then on a surface of the second degree, and the h chords form 

 the complete intersection of two cones. The number of conditions is, 

 if n = 2t, i^+ 2t + 9, or f^ + 2.' + 8, according as < is > or = 2 ; and if 

 n :^ 2t + 1, it is t- + 3t + 10, or t^ + 3t + 8, according as ^ is > or = 1.' 



I repeat that I have not yet been able to find, for every such system, 

 of /* rays as that constructed in Ai^t. 4, a cone which has them for double 

 edges. 



6. By considering the monoids passing through a curve [«, /(], I 

 have arrived at the following theorem : 



The cone of lowest order passing through the h chords is of the 

 degree n—j, j being defined by : Ji/ I> A > ^', + i. Hence it follows that 

 for a surface of the order m > n ~j to contain the entire curve will be 

 a condition of the following degree of multiplicity : - 



mn + 1 — {\{n — l)(ii — 2) — h] = mn + 1 — ^. 



Perhaps the inferior limit of m may be smaller, but certainly the 

 theorem does not subsist in all cases. 



7. I have approached the theory of twisted curves in another manner, 

 by inquiring into the curves situated on a surface of the third order, — my 

 old field of research. Following the example of Pliicker, Cayley, and 

 Chasles, who have considered the curves on a surface of the second 

 degree,^ I construct all curves on a given cubic surface by means of re- 

 lations between the pencils of planes Avhose axes are two right lines of the 

 surface, not situated in the same plane ; in this manner I find 3, 6, 7, 

 11, 14 species of curves of the orders 5, 6, 7, 8, 9 respectively; on 

 these I will make but two obser?ations. 



First, it often happens that two species agree in their order n as well 

 as in the number h of their apparent double points, and hence also in 

 their rank r, and in the degree of their manifolduess on the surface (which 

 always is ^r), but differ in the number of their intersections with the 

 right lines of the surface. For instance, there are two curves of the 

 sixth order with 10 apparent double points, and with the degree 5 of 

 manifoldness on the surface : one of them meets 6 right lines 4 times, 15 

 twice, 6 not at all ; the other meets 1, 1, 10, 5, 5, 5 lines, respectively, in 

 5, 4, 3, 2, 1, points. 



The second, and, in my estimation, the more important observation is, 

 that most of these curves on a cubic sui'face are not general ones of 

 their kind, that is to say, if we construct, on a cubic surface, all possible 

 curves \_n, K\, and then construct all possible cubic surfaces, we shall 

 not reach the degree of manifoldness which is proper to a curve [«, li\. 

 For instance, the cnrves [9, 28] must have the degree 36 of manifoldness ; 

 but those situated on cubic surfaces have only the degree 27. 



' Kohn also seeks these numbers, but makes an error in the first case. 



- Ed. Weyr, in his pajDers on twisted curves of the .sixth order (Comjftes 

 Bendus, vol. Ixxvi.), appears to have overlooked the circumstance, that this con- 

 dition is not necessarily of the degree mn + 1, and hence to have arrived at 

 conclusions which are not all valid. So far as these curves are concerned, I might 

 refer him to my book on Surfaces of ilie Third Order, Art. 73. 



^ Plucker, Crcllv'is Journal, vol. xxxiv. ; Cayley, Phil. Mag. July 1861 ; Chasles, 

 Comjjtes Rendus, vol. sliii. 



