1186 



determined bv Joi.ibois are indicated, who, as also follows from the 

 table, has only been able to continue his research up to 312°. 



In a subsequent communication we shall give some theoretical 

 considerations in connection with the results stated here, and also 

 discuss the vapour pressure line of the solid modification, which 

 we determined accurately already some time ago. 



Anorg. Chein. Laboratory of the University. 



Amsterdam, March 27, 1914. 



Mathematics. — "A bilinear congruence of rational tiuisted quartics.'' 

 By Professor Jan de Vries. 



(Gomraunicated in the meeting of March 28, 1914). 



1. The base-curves of the pencils of cubic surfaces contained in 

 a net [*l*^^ form a bilinear congruence. ') 



If all the surfaces of the net have a twisted curve {,'" of genus 

 one in common, and moreover pass through two fixed points ^i,i/j, 

 every two <ï»' cut each other moreover along a rational curve q*, 

 which rests on q^ in 10 points, ^j 



A third 0' cuts i/ in 12 points, of which 10 lie on ^)' ; the 

 remaining 2 are H^ and H„. Through an arbitrary point P passes 

 one c/ ; if P is chosen on a trisecant t of q% then all «ï»" passing 

 through F contain the line t, and q' is replaced by the figure com- 

 posed of t and a r', which cuts it, and meets q^ in 7 points. 



2. In order to determine the order of the ruled surface of the 

 trisecants t, we observe that each point of (/ bears two trisecants, 

 so that ^" is nodal curve of the ruled surface {t). We can now 

 prove that a bisecant b, outside q\ cuts only one trisecant, from 

 which it ensues that (t) must be of order five. 



The bisecants b, which rest on the bisecant b^, form a ruled sur- 

 face (b) of order 7, on which />„ is a (piadruple' line. In a plane 

 passing through b^ lie three bisecants ; as to each of those three lines 

 the point of intersection of the other two may be associated, by 

 which a correspondence (1,1) is brought about between the lines b 

 and the points of (/, {b) is of genus one. A plane section of {b) has 



1). See my communication in these Proceedings, volume XVI, p. 733. There 

 I have considered the case that all <t>^ have in common a twisted curve i/> of genus 

 two, so that a bihnear congruence of elliptic quartics is formed. 



~). See e.g. Stukm, Sijnthetische Untersuchungen iiber Flaclien drltier Ordimng 

 (p.p. 215 and 233). 



