(48) 



fke ciirrc 'm I jnunfs //, //k'ii II alisDi-lis r ftrupcr jiotiils i)f coiihu-l 

 ivilh IdiKjcnls from /* irln'ii P hes oii I niilsidc //, and I -\- r nj-oncr 

 po/iifs of coiifdct ii'hcii /•* coinci<l('s irith II. ') 



Now, if >S' is siicli a poiiil, where ;ill l)r<iiielies Iiavc (Ik^ same 

 tang-ent S l\ llien m z= t. -\- i\ whilst aeeording (o llie ahoxe tlieoi-eni 

 S eounts tbr rz=zir--t points ot'eoiilacl with tangents IVoin l\ II" besides 

 the hi-anehes tonching >S7'* still more hi'anches pass throngh X then 

 these latter do nol give rise to any new points of' contact coinciding 

 with >S'; they canse however the same increase of the nnnibers / and 

 10, so that they leave ir — / nnchanged. Then too lo — / represents the 

 nnniber of points of contact, which in consecpience of the singnlar 

 sitnation of P coincide with S. 



If /■* is not situated on one of the tangents in S, then //' = /, so 



1) For a branch which can bo ropresontcd by one single PnsEi'x-devclopnionl 

 this theorem can be proved i. a. out of the relation existing betweeu the 

 developments in point- and in line-coordinates. By addition follows immediatelv 

 the same theorem for more branches having the same tangeiit. In a paper 

 entitled : "An equation of reality for real aiid imaginary plane curves with higher 

 singularities" (These rroccedings of April 23'''' 1904, p. 764) I made use of the 

 same theorem (p. 765) and referred for the deduction to Stolz, Zeuthen and 

 Stephen Smith. I omitted however to mention G. Halphen, "Mémoire sur les 

 points singuliers des courbes algébriques planes", Mémoires pré?, par dirers 

 sarcnits a VAcadnnie dos Sciences (2), t. 26, (1<S79), n". 2 (112 p.). This extensive 

 paper was already ottered to the Paris Academy in April 1874 (see : Comptes 

 Revdvs de F Académie des Sciences de Parif<, t. 78, p. 1100—1108, where the 

 aulor communicates some of his results) so that this paper has the priority. 

 Halphen formulates the theorem somewhat differently, namely (I. c. Théorème III, 

 p. 42 or Tl'iéorème II, p. 50): 



Théorème. Ja) somnie des ordres des cdii/acfs^ dei< l)runclies d'niie co/ir/x' arcc 

 line de ses tangentes est egale a hi miiHiplicité dn point correspondant a cette 

 tavgente dans la conrhe corrélatire. 



The relation between the developments of a branch in |)oinl- and in line- 

 coordinates was first considered by A, Cayle> ("On the Highci- Singularities of a 

 Plane Curve". Quart. Joiirn. of Math., Vol. 7, (1S(;6), p. 212, Collected Math. 

 Pap., Vol. '), p. 520 or "Note sur les singularilés supérieures des courbes planes" 

 Crelle:^ Journal, Bd. 04, (1S65), p. 860, Colt. Math. Pap., Vol. n, p. 424). If 

 y = Axi^-{- B x'i -\- . . . .(p "> \) is the development in point-coordinates then 



V q 



Cavi.ev gives for the devek)pment in line-coordinates Z = A' A'/'—' + ^' ^ '""' + •••, 



■ ^ P ~\~ 1^ 'I ~\- ■ ■ • \ 

 where llie ucneral lorni u\ llie exponents is . : liere A, u,.... are posi- 



live integers. hnpli<'itly llie llAM'UKN-liieorcin is iiicludcd in this. (.Iaylev, however, 

 does not enter farther into the relation l)elween the developments and i\w^ not 

 state Ihe theorem. 



Let me finally notice, that the Iheoreni has also been stated by M. Nötheh, 

 "Uebor die singuliiren Werlhsysleme einer algebraischen Function nnd die singuliiren 

 Punkle einer algebraischen Curve", Math. Annalen, Bd. 9, (187G), p. 166 (sp. p. 182). 



