( 31 ) 



p = (3?i' — 6?i + 2) (?i — 3). Farther more q=z2{n~- 3) (n — 1) (2??,— 1), 

 whilst tlie number of right lines PQ i-esting on an axis is ofeonrse 

 equal to 3/i {n — 2) (?i - 3). So ^ = '2 {n — 3) (2/i' — 3;^ + 2) ; this 

 is also the number of four-pointed tangents through a given point <(. 



The number of right lines t^ in a given plane is equal to tlie 

 number of points of undulation on the curves c" of a pencil; tliis 

 number I have determined in a {^receding paper ^). 



TJie four-pointed tangents foim a comjruence of order 



9 



2{n — ^){2n' — 2>n-\-2) and of class — {n—3){^n'-^n'—Sn~^^). 



Li 



4. If we wish to apply the abo\e-mentioned formula of coincidence 

 to the pairs of points of intersection Q,Q on the right lines t^ through Z 

 we have to substitute p z= q=z2{n — 3) {n — J) {In — - 1) {n — 4) and 

 g^=zZn{n — 2) (n — 3) (?i — 4). For each point Q belongs to {n — 4) 

 pairs and each right line /, bears [n — 3) (/i — 4) pairs. We then 

 find f =: {n — 3) {n — 4) (5m' — Qn -\- 4), i, e. the number of tangents 

 t^o through the point Z. 



In the above-mentioned paper I have determined the number of 

 right lines having with a curve of a pencil (c") a three-pointed iind 

 at the same time a two-pointed contact. 



The tiiw-tkree-pointed tangents form a congruence of order 

 {n — 3) {71 — 4) (5?i' — 6/1 -(- 4) and of class 



— {n - 4) {n — 3)' {lOn* + 35n* — 21?i' — SOn -f 20). 



5. Each principal tangent t^ ha\ing its point of osculation in a 

 point S of the base-curve bears still {u — 3) points of intersection Q 

 with the surface i^" osculated by it. As S is point of contact of 11 

 four-pointed tangents the locus of the points Q will have an eleven- 

 fold point in ^S. As an arbitrary plane through S evidentl}' contains 

 3 {71 — 3) points Q (§ 1) the order of the curve {Q) is equal to (3n -f 2). 



When applying the formula e =^ p -\- q — g to the pairs Q, Q' 

 which the cubic cone with vertex S bears, we have to put 

 pz=q = (3?i -f 2) {n — 4) and (/ = 3 (?z — 3) {n — 4). Then ^ve get 

 8 := (?z — 4) (3?i -f- 13). So this is the number of tangents ^3,2 for 

 which the point of osculation lies in S. 



In other words, o is an {71 — 4) (3?/ -{- 13)-fold curve on the locus 

 [7i,] of the points of osculatio7i of tangents tz^ to surfaces of {F"). 

 Now the points of osculation of the right lines 1^,2 of an F" form a 



1) "On linear systems of algebraic plane curves", (Proceedings, April 22, 1905.) 



