( 820 ) 



Because each of the In (n — 2) (n — 3) double tangents out of 

 represents 2 (?? — 4) coincidences d^iw, the number of coincidences 

 J):^ 11' is represented by 



(„ _ 4) („ _ 3) (2m' + 5« — 6) + (w — 4) {?* — 3) (on" -\- 5// — 6) — 

 — 4 (n — 4) (n — 3) (m — 2) w z= 3 (n — 4) (« — 3) («' + 6« — 4). 



In a pencil (c") n.'e find that 3 (« — 4) {n — 3) {n'' -\- Gn — 4) curves 

 have an inflection, of which the tangent touches the curve in one other 

 2)oint more. 



In the paper quoted above I tiiouglit I was able to determine this 

 number out of the points of intersection of the curves (i>)and(]F); 

 here I overlooked the fact that a point of contact of a double tangent 

 can be tangential point IF of another double tangent. 



§ 5. To find the number of tlireefold tangents 1 consider the 

 correspondence between the rays projecting out of <) two points 

 IF and W' lying on the same double tangent. The characterizing 

 number of this symmetric correspondence is evidently equal to 

 \ (n — 4) (n — 3) {5n^ -\- 5n — 6) (n — 5), whilst each double tangent 

 borne by O replaces 2n {n — 2) {n — 3) {n — 4) (n — 5) coincidences. 

 The number of coincidences W i£2 IC amounts thus to 



(h — 5) {n — 4) {n — 3) {an' + 5n — 6 — 2n' + 4:?i). 



As each threefold tangent bears three of these coincidences we 

 have the property : 



Li a pencil (c") ive find that {n — 5) (n — 4) (n — 3) [n'' -\- 3n — 2) 

 curves have a threefold tangent. 



§ 6. In my paper indicated above I have tried to detei'mine the 

 number of undulation-points out of the points of intersection of the 

 intlectional curve (/) with the locus of the points ( 1") which c" 

 determines on its inflectional tangents. As each inflection which is 

 also tangential point of another inflection is common to (/) and 

 ( V), the number found elsewhere is too large. The exact number I 

 can determine by means of the correspondence between the rays 01 

 and OV. 



As the orders of (/) and ( V) are a{n — 1) and 3(?i — 3){n''-}-2n — 2) 

 and each of the 3n. (« — 2) inflectional tangents drawn from O replaces 

 (n — 3) rays of coincidence, we get for the number of coijiciilences 

 ƒ= V 

 6(«— 1) (n — 3) + 3(w — 3) (n' + 2?t — 2) — 3w(«— 2) (w — 3) = 6^«-3) (Sn—2). 



In a pencil (c") we fni.d that 6 (/i — 3) (3 n — 2j curves have a 

 four-point tangent. 



