( 42 ) 



Mathematics. "(^n an I'.i'ju-i'sslon f of the class of an dhjehrdic 



plane carrc icilh li'iijlicr siiKiiiJarltles." By Mr. Frkd. Schih. 

 ((V)miniini('atod by Prof. 0. .1. Kortkwf.g.) 



If an akichraic jilanc ciwre is given hy an equation in Cartesian 

 iwh/t-coorcfinatcs, its tvdcr n can l)e immediately read from the 

 equation. The class k of the curve is best defined as the order of 

 the equation in line-coordinates. However, in the following it is 

 my intention to I'estrict myself exclusively to point-coordinates and 

 then the class can l)c dclined as the number of morahle points 

 of intersection of the ciii've with the first polar or as the number 

 of proper tangents to be (h-a\vn from an arhitranj point P to 

 the curve. To obtain e.vclusively different points of contact not 

 situated in u^anifold jwints of the curre we nuist understand by an 

 a)'ln'tr(tri/ point a point that 



l^f does not lie on the curve, 



2'''^ does not lie on one of the tangents in a nntni/old point of 

 the cui've. 



3"^ does iu)t lie on a tangent in a luiifold |)oint having with the 

 curve a contact of a higher order than the first. 



A wanifold point of the ciii-\o is a curvepoint w liich cannot 

 be a single point of inteisectioii with a straight line. The lines 

 connecting P Avitli the manifohl points must not be counted as 

 pro})er tangents. 



From the above mentioned definition of tlie class another one can 

 be deduced, whei'c no single resti'iction is made witli respect to the 

 situation of tiie point /■*, which thus holds good for any |>oint J*. 



To begin with, we make the restriction that P nnri/ not lie on the 

 corn'. Suppose P to lie on the tangent in a point N of the curve, 

 where N may be a manifold point or a single point with a tangent 

 intersecting in more than two points. If the straight line /*/S' cuts the 

 curve in n' coinciding points .S', whilst an arbitrary straight line through 

 ,S^ cuts the curve in t coinciding ])oints S, then .S' counts for 2/;— /of 

 the k proper points of contact with tangents from F to the curve, 

 in other words //' — t [)oints of contact approach >S' when /^approaches 

 the tangent in S. If is of no importance whether the cur\e has 

 one or more branches through S, touching SP, neither whether the 

 curve has branches through .S' not touching SP or not. 



The above mentioned follows immediately from the following: 



Theorem. Let R he a point of an algebraic cun^e ivhere all branches 

 thronc/h P hare the same tanc/oit I which intersects the curre in /-j-z' 

 coinciding points P, luhilst every other straight line through R intersects 



