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Mathematics. — "^l// ('(/uafion of reaJllij for rctd m/d inmt/iittn'i/ 

 jddne cm-res iritli li'KjIu'r s'ui<iiil<iritles'\ \\\ Mr. Fukd. Schuh. 

 (Communicated liv Fiof. 1). J. Kortkwe^.) 



((A)mmunifaleil in the meeting of Maicli I'J, 1904). 



For a plane algebraic curve having an equation with real coeflli- 

 cients only and possessing no higher singularities than the foiu- of 

 Plücker, Klein ^) has deduced (as an extension of relations of reality 

 found by Zeuthen in a CV) the equation 



» 4- j3' + 2r" = /- + x' + 2 d" (1) 



where 



n is the order, I- the class of the curve, 



^ the number of real inflexions, 



y.' the number of real cusps, 



t" the number of real isolated bitangents and 



d" the number of real isolated double points. 



This equation of Klkin' can l)e extended to curves with /ii(//ier 

 shuiuJarities and it then becomes most remarkably simpler and 

 hnrn'iahh/ Itohls good also for curres m vdiose equation ima(/inari/ 

 coeffcients appear, which is not the case with the ecpiation of Klein. 



The equation found I)v me runs as foUou s : 



n-\-:£' r,^k-^r2:'t, (2y 



Here ^'t^ <lenotes the sum of the orders of the singularities with 

 real imnt, 2' i\ the sum of the classes of the singularities \\'\{\\ real 

 tane/ent. By an element of the curve I understand in the following a 

 point of the curve together with the tane/ent belonging to it, exclu- 

 sively as forming a part of one branch of the cur\e, which can 

 be represented by one single PiisEux-development (\\'itli exponents 

 fractional or not). The element I call a singvlarity : 1^^ if point or 

 tangent or both are singular, or 2"<^ if point or tangent or both 

 belong to several elements of the curve, or 3'^^ if the point is real 

 and the tangent imaginary or reversely. 



If through the same point more branches pass, we call the point 

 a manifold singular point ; this is to be regarded so many times as 

 a singularity as there are branches passing through it. Correlative 

 to this is a manifold singular tangent. 



1) F, Kleix, Eine none Relation zwischen den Singularitaten einor algehraischen 

 Curve, Math. Ann., Bd. 10 (1876), p. 199. 



