( 765 ) 



With Plücker M we nnderstand by I lie order t of a single sin- 

 gularity the nninbei* of the points of intersection coinciding in the 

 (singnhir) point with an arbid'ary hue of irifersection through that 

 point. If eventually more branches [»ass Ibrough Ihe singular point, 

 we must of course count lliose points of intersection only, which l)y 

 a slight displacement of the intersecting Hue are found on the 

 branch belonging to the singularity in (piestion. (correlative to the 

 order is the chtss r of the singularity. This class is at tiie same 

 time equal to the number of points of intersection ap|)roaciiing the 

 (singular) point along the branch in question witii an intersecting 

 line passing through that ))oint and al)out to coincide with the (sin- 

 gular) tangent. 



If we regard this last quality as the definition of the class of 

 a singularity, tlien the development in point coonhnates with the 

 (singular) point as origin ajid the (singular) tangent as axis of -/; 



t-\-0 



becomes y =: a ,r: -|- . . . . , after having given all exponents an 

 equal denominator though as small as possible (for a small value 

 of .li gives t small roots //,. o]i the contrary a small value of // 

 gives I -j- V small roots ,*;). If farthermore ^Y aiul Y are tiie line 

 coordiiiates of the straight line y -j- X .c -j- }"= 0, then the development 



in line coordinates becomes Y=zAJ( " -{-... '^), from which ensues 

 the correlativeness of / and i\ At the same time it is evident from 

 this that order and class of a singularity can l)e read immediately 

 from the corresponding development, from those in pointcoordinates 

 as well as from those in Hue coordinates. 



In (2) ^'/j denote.'^ a .miuinatioii frith i-espect to the suuiuhtrltles 

 witJi retd point, 2'v^to the .siiKjidarities iridt real t<tn,(/ e n,t, whe.vG 

 a manifold singularity must be taken into consideration as many times 

 as it possesses single singularities. Ilei-e not only tiie higher and 

 PLÜCKER-singularities must be counted, but also those elements of 

 the curve, the counting of which has an influence on the ecpuition 

 (2); thus also those elements {t = r =: ^) of which the point is real 

 and the tangent imaginai-y or reversely. It is clearly inditterent 



Ï) J. Plücker. Theorie der algebraischen Curven. Bonn, A. MAiicuri, 1839, p. 205. 



~) 0. SroLZ. Ueber die singulüren Punkte der algeljraisclien Functiouen und 

 Curven. Math. Ann., Bd. 8 (1875), p. 415 (spec. p. 441—442). 



H. G. Zeuthen. Note sur les singularités des courbes planes. Math. Ann. Bd. 

 10, p. 210 (spec. p. 211—212). 



H. J. Stephen Smith. On the Higher Singularities ofplane Curves. Proc. London 

 Math, Soc, Vol G (1874-75), p. 153 (spec. p. 163-1G4). 



