( 498 ) 



oyi = q.K'^q.K' + qzK' ^ (z = o, i, 2, 3, 4) ... (4) 



hold, and now the equation sought for is found by eliminating the 

 nine quantities ).^,).^,).^, Pi,p^,2h^ ^/i? ?2j ^3 ^^^^ of the ten equations 

 (3) and (4). Tins takes place bj inserting the values given by (3) and (4) 

 in the left hand member of the second equation (2). For by this we find 



.'6'g X^ X^ A' 3 



A' J A' 2 X^ X^ 



2/0 III V, Ih 



Vi z/2 :'/3 Va 



1 1 



Pi V2 Pz 



Px^i PJ'^ PzK 



Ql 9-2 ?3 



QiK q-iK Qz^3 



= 0. 



''1 ''2 '"3 



We considered in the above cited communication equations forming 

 the extension of tiie first of the equations (2) to the curve k-" of 

 the space Sp2n- In connection witli this we shall notice that the second 

 of the equations (2) admits of corresponding extensions, in which 

 those of the first are included. However, these will be developed 

 elsewhere. 



Mathematics. — "The Plückei{ equivalents of a cyclic point of a 

 tmisted curve.'' Bj^ W. A. Versluys. (Communicated hj Prof. 



P. H. SCHOUTE. 



If a twisted curve C admits of a higher singularity (cyclic point) 

 of order n, of rank r and of class m, it is to be represented accord- 

 ing to H ALPHEN ^) in the vicinity of this singular point M by the 

 following developments in series : 



y = t-+r [t], 



Z = P'+r-ï-m |_^], 



where [i] represents an arbitrary power series of t, starting; with a 

 constant term. 



If 71, r and jn satisfy the conditions that 



1" n and r, 



2" r and m, 



3" n and r-\-'m, 



4° n-\-r and m 



are mutually prime, then this higher singularity M (71, r, m) for 



(-4) 



'1 Bull. d. I. Soc. Mat. d. Fiance t. VI p. 10. 



