( 30 ) 



of intersection of tlie indiccitcl tigure with the riglit line z^ = 0, 

 z^ = 0. Substitution in (2) and in (3) and elimination of z.^ and z.^ 

 gives an equation containing the coefficients of (2) and (3) succes- 

 sively in the orders 4 and 3. Hence the resultant in the coordinates 

 ?/ is 'of order 4 {2n — 3) -f 3 {%i — 4) oi' 14/i — 24. 



So the number of points of intersectioji is %f {In — 12). 



Applying the same treatment to (4) I find for the order of (.y) 

 the well-known number %i^ {n — 1). 



The foitr-jwinted tam/ents havlmj their points of contact on the 

 base-curve o form a scroll of order In" {Qn — 11), on which ö is 

 elevenfold. 



For n = 3 this scroll passes into the locus of the right lines on 

 the surfaces of a i)cncil {F^), thus ijito the scroll of the trisecants 

 of o' which is of order 42. For %t[%n — \\) we now find 126, 

 correspondijig to tiie fact, Ihat each trisccant does duty as right line 

 t^ for three points S. 



On each surface 7^" the points of contact V^ of four-pointed 

 tangents t^ form a curve of order n{\\n — 24). As a is evidently 

 an elevenfold curve of the locus of the points of contact P^ belonging 

 to the P" of the pencil this locus has in common with every i^" a 

 locus of order n (ll>^ — 24) -\- \hv . 



The points of contact of four-pointed tangents form a surface of 

 order 2 (lln — 12). 



For ?z = 3 we find the scroll of trisecants of order 42. 



3. As the tangents t^ passing through a point Z form a cone of 

 order ^n[n — 2) and two princi[>al tangents have their point of 

 contact in Z the locus of the points of contact 1\ of the right lines 

 ^3 containing Z is a curve of order 3?i(/i — 2) + 2 with double point 

 in Z. 



Each of those right lines t^ cuts the surface P« osculated by it in 

 (,i _ 3) points Q more. The locus of the points Q is a curve of 

 order 37i(?i — 2)02 — 3) -j- ^z^ + 2)(n — 3) or 2{n — Z){n—l){2n — l). 

 For, through Z pass {if + 2) {ii — 3) tangents t^, osculating the surface 

 indicated by Z in an other point ^). 



To find the number « of the coincidences of P^ with Q, I make 

 use of the well-known formula 



€ =r 2^ 4- ^ — .(7, 



which appears when the pairs of points I\ Q are projected by a 

 pencil of planes. Each point F belonging to (vz— 3) pairs we have 



1) See Cremona— GuRTZE, Oberflachen, p. 66. 



