( 364 ) 



greatest in the most distant elements and greater in Br -|- I than in 

 Br 4- CI. 



From the researches of Moissan and others it follows that Fluorine 

 yields the compound IF^ which is stable even in the vapour-condition. 

 With Bromine, the compound BrFg is formed but no compound is 

 formed with Chlorine. This, also, is in harmony with the above result. 



As, however, the compounds with Fluorine have not been studied 

 from the standpoint of the phase-doctrine, there does not exist as yet a 

 reasonable certainty as to their number or their stability. 



Mathematics. — "Second communication on the Plücker equivalents 

 of a cyclic point of a twisted curve." By Dr. W. A. VERSiiUYS. 

 (Communicated by Prof. P. H. Schoute). 



§ 1. If the origin of coordinates is a cyclic point [n, r, m) of a 

 twisted curve C the coordinates of a point of C lying in the vicinity 

 of the origin on a branch passing through the origin can be repre- 

 sented as follows: 

 A* = a <", 



y=.h^ «"+'• + h^ v^+r+\ -f b^ ^«+'•+2 -{- etc., 

 e = Co <«+'+"> -f- Cj «»+'+'»+i -f Cj «"+'-+»»+2 J\- etc. 

 Let g-i be the greatest common divisor of n and r, let q^ be that 

 of r and m, q^ that of m and n-\-r and finally q^ that of n and 

 r -j- m. 



If q^ = q^ =: q^ = q^ = 1 the PLtJCKER equivalents depend only 

 on n, T and m. In a preceding communication ^) I gave the Plücker 

 equivalents for this special case "). 



§ 2. If the 4 G. C. Divisors q are not all unity, the PLtJCKER 

 equivalents of the cyclic point (?i, ?', ni) depend on the values of the 

 coefficients h and c, just as in general for a cyclic point of a plane 

 curve given by the developments: 



X = V\ 



y =: t^+^ -\- d^ i«+'«+i -f fZj i«+"»+2 -}- etc., 

 the vanishing of coefficients d influences the number of nodal points 

 and double tangents equivalent to the cyclic point [n, m) '). 



1) Proceedings Royal Acad. Amsterdam, Nov. 1905. 



") The deduction of these equivalents is to be found among others in my treatise : 

 "Points sing, des courbes gaudies données par les equations: x = t>', y — tn-\-r^ 

 2 = ^1+'-+'"," inserted in "Archives du Musee Teyler'\ série II, t. X, 1906. 



3) A. Brill and M. Noether. Die Entwicklung der Theorie der algebraischen 

 Functionen, p. 400. Jahresbericht der Deutschen Mathematiker-Vereinigung, 111, 

 1892—93. 



