( 494 ) 



exact number; so we also possess the exact numbers of the double 

 tangents and the inflectional ones, so that only those of the double 

 points and cusps are missing. The Pltjcker formula i — x = 3 (r — ii) 

 furnishes us with x = e -|- 3 (,u — r) : if we introduce the values, we 

 find X = 6 n* — 26 n' -[- 24 ji' + 32 tz — 36. Finally the formula 

 V = (I (a — 1) — 2.'/ — 3x furnishes us with the double number of double 

 points: 2(f=n{n — 1) — V — 3x, hence: 



2d= (27i^— 9;z'+10/i^-f 10/i— 12) (27z^— 9/i' + 10/i^+10/i— 13) — 



— 2n (n—A) - 3 (6?i^—26?z'4-24?z*-f 32/1—36). 



Summing up we have thus found : the apparent circuit of £2 on 

 an arbitrary plane is a curve of order 2n* — ^n^ -\-\0n- -\-10n — 12, 

 of class 2n {n — 1), ivith a nw.nber of double points = é (see above), 

 a number of cusps = x (see above), 2üith {n-\-l) (ji) [ii — 2) (p. — 3) 

 double tangents, the projections of the double generatrices of £2, with 

 n {n^ — 4) inflectional tangents, the projections of the cuspidal edges 

 of i2, and irith tiro n{n — l)-fold tangents, the projections of the two 

 directors r^ and r.^ . 



§ 9. If i2 is really a conoid, i. e. if r, is the line at infinity of 

 a director plane, then as a rule the latter is chosen as plane of pro- 

 jecJion, and so the projection of the surface on a plane througli one 

 of the two directors becomes of importance. In the numbers men- 

 tioned at the end of the preceding ^ no change takes place; so in 

 the case of the ccmoid the apparen<. circuit on a director plane 

 possesses ?i (n— 1) parabolic branches. It is a different thing, however, 

 if the conoid is a right one, i. e. if r^ is normal to the director plane; 

 if then the latter is horizontal, and if the apparent circuit of .0 is 

 required for the point Zx, as centre, then we have to project out 

 of a point of the surface itself, and that one lying on the n {n — 1) 

 fold line r^. It is now immediately clear that the apparent circuit is 

 entirely modified; fur a line through Z^, cuts 52 besides Zx only 

 in n{n — 1) points, and only the generatrices passing through these 

 points give rise, when projected out of Z«, to tangents of the apparent 

 circuit; however, they all pass in projection through the point of 

 intersection R^ of r^ with the director plane, from which ensues that 

 the pencil round R^ is discarded and that ?i {n — 1) times. 



The plane through Zx> and one of the n {n — 1) generatrices of i2 

 (lying entirely at infinity) is indefinite, i. e. each suchlike plane is a 

 tangential plane through ^ao; of the apparent circuit we have to 

 discard n {n — 1) pencils whose vertices are the points of intersection 

 of the generatrices through Za, with r^^. These pencils and those 



