( 620 ) 



il ;i[)|)e;irs oii llio foeal siii-face of ;i plaiiu or twisted curve toiu'lihi^ 

 tlie plane at iiiliiiil}'. 



So the fornmla of Crkmona becomes : 



.v(o-- 2)= iinmber of times that s meets the curve 

 of contact of yl -f 3 ;. + P. 



Tiie curve of contact of .1 consists of tlie generating lines of 0' 

 of wiiich the tangent pUxne passes througli A. Of these generating 

 lines two are situated on the cone K, each of these meeting the 

 curve .s- three times and two are situated on O^ as .1 lies on the 

 osculating plane T; the latter meet .v each one time. This number 

 of i)oints of intersecticHi of x witii Z." O' is tiius eight. According 

 to Crkmona they must eacii be counted one time, so the first term is tS. 



The section of O' with [" consists of d., and of the tangent in P 

 to (/.^, both counted double. So the second jtoiar curve of A for this 

 curve or the section of I" with jL'^ W consists of a cui-ve of order 

 four, touching d.^ twice in ]\ having thus with (/,, and with s 

 too only four points in common. The formula of Ckkmona gives: 



G X 4 = 8 + 3 X ^. + 4. 



Consequently : P. = 4. 



The points ). are the points outside P where the cuspidal curve 

 meets another sheet. As (\ meets the cone K in all six times, 6', 

 can meet the cone K in P only twice, so ('^ has also with the 

 curve d^ lybig ^'^^ ^^' "*'•* ^^^'** points in common. 



§ 7. If W'e had taken A outside the plane T" this change would 

 have had no influence on the number of j)oints )., but the term 

 issuing from the lines of contact would ha\e l)een one more, so P 

 would in that case have counted for 3. 



By a]3plying the above used method to the nodal curve d^, we 

 find that A'^ 0' of an arbitrary point A lying outside V also meets 

 t/j in three points. The singularit}^ P counts for six points of inter- 

 section of the total nodal curve | with A* 0\ thus for as many 

 points of intersection as tw^o points P.. 



Out of the formula of (^RK:sroNA for r (o — 2) it is evident that P 

 counts for two points of intersection of the cuspidal curve 6'^ with 

 A-* ()' , thus also for two points P., From what precedes ensiies that 

 the cone K passing through d.^ is an ordinary touching cone and 

 that the point of contact P counts in both formulae of Cremona for 

 two points k. 



§ 8. Let tlie curve ^' be a general twisted curve of order i\ Let 

 O be its (h'vclopabh', the class ;*, the rank (* and let the order of 



