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lines of a pencil iinfh J' as verte.c in n movable points, which distribute 

 themselves for the I'arions .straijht lines of the pencil over N^, N^ ... . 

 branches 0} the curve, then i^n' -\- h 2 {n'—JSl,), (where :S (n'—X,) 

 is taken over all the straight lines through Pj, has for every point P 

 the same value equal to the genus of the curve. 



This theorem liowever can be considerably extended by nuiiving 

 use of the property that tiie gemis ot" the curve does not change 

 by a one-to-one transformation. If namely we apply to the whole 

 figure a rKK:MONA-transibrmation, then // remains the same, but n' too. 

 The straight lines of the pencil arc liii'iied by 1 lie transformation into 

 rational curves having multi|)le points in the fundamental |)oints of 

 the transformation. Tiirongh every i)oint lying outside those fundamental 

 points only oite of the rational curves passes, so that we treat a pencil 

 of rational curves; the manifold points here making the ciirxes to 

 rational cur\es are nol present as movable points but as fixed 

 points, which gives rise to linear relations between tlie coefficients 

 of the curves. The movable points of intersection with a straight line 

 are now transformed into movable points of intersection with a rational 

 curve, and tiiey remain the same in number on accoujit of llie one- 

 to-one character of the transformation. 



Hv a CKKMONA-transformation a branch is I'liithcrmore always 

 transformed into one single branch (where we always understand by 

 a /iraoch the whole (»f the curvepoints whose coordinates satisfy 

 the same PiisEUX-developmenl 1. If thus the u.' movable points 

 of intersection with a straight line distribute themselves over iV 

 branches then in the transformed iigure the n' movable points of 

 intersectioii \\illi the rational curve originated by trausfoniiation of 

 the straight line distribute themselves also over .V l)ranches. 



From this it is evident that all quantities of (Mpialion (2) are 

 invariant with respect to rational transformation, so llial /hr r(j nation 

 (2) holds good unchanged if the pencil of straight lines is irplaced 

 by a pencil of rational curves. 



This gives rise to the following theorem: 



Theorem II. If an algebraic phine curve is intersected by a pencil 

 of rational curves in n' movable points distributing themselves for the 



different curves of the pencil over X,, X^, branches of the fixed 



curve, then 1 — n' + i 2 {n' — Xj, (where 2 (n' — Xj is taken 

 over all curves of the pencil), has for every pencil of rational curves 

 the .mme value equal to the genus of the feed curve. 



This theorem can then be extended from a pencil of rational curves 

 to an ai'bitrary algebraic pencil of curves by means of the following 

 considerations which are however less rigorous than the preceding ones. 



