( 493 ) 



order of llie projecting cone out of O or of tiie apparent circnit on 

 a plane, or the class of a plane section of ii. 



For the class of the api)arent circnit we must know the number 

 of tangents through an arbitrary point P of the plane of projection. 

 Now OP cuts the surface i2 in In {n — 1) points; through each of 

 these passes a generati'ix, and the |)lane through these and OP is 

 a tangential plane through OP, so the trace of that plane is a 

 tangent to the appaj-ent circuit ; the class of the apparent circuit is 

 therefore 2n {n — 1). 



Let us bring a [)lane through and a torsal line whose cusp lies 

 on )\. It cuts SI according to a curve of order 2n (n — 1) — 1, and 

 as the complete intersection, consisting of this curve and i\ , 

 must have n (n — l)-fold points on i\ and )\, the curve itself 

 has on the directors \n {a — i) — Ij-fold points. These points lie at 

 the same time on the generatrix; the only still missing point of 

 intersection with this generatrix coincides with the cusp, and in 

 projection the apparent circuit touches in this point the torsal line. 



A plane through and a double edge of SI contains as residual 

 section only a curve of order 2n {n — 1) — 2 with \n [n — 1) — 21- 

 fold points on i\ and r,, and which thus cuts the double edge in 

 two points more; a plane through a double edge is therefore a double 

 tangential plane and the two points just mentioned are the points 

 of contact. The projection of the double edge is a double tangent 

 of the apparent circuit; the points of contact are the projections of 

 the two points just mentioned on SI. 



In a plane through and a cuspidal edge the latter counts 

 likewise for two, so that here too remains a residual section of 

 order 2n {n — 1) — 2 with \n {n — 1) — 2|-fold points on i\ and i\ ; 

 the two missing points of intersection with the cuspidal-edge coincide 

 here and the projection of this edge becomes an intlectional tangent 

 of the apparent circuit. 



Let us now imagine a plane through and )\. Let >S', be the 

 point of intersection of this [)lane with y.,, then to this point correspond 

 n{n — 1) points on i\, and the projection of i\ touches the apparent 

 circuit in the projections of those points; the apparent circuit has 

 therefore the projections of ;■, and i\ as n{;n — l)-fold tangents. If 

 we now reduce these mulliple tangents to double ones and if we 

 then suppose that the double tangents and the intlectiojial tangents 

 just now found are the only ones that the curve possesses, and if linally 

 we remember that the class of the curve is 2«(?z — 1), then the Plücker 

 formula to determine the order becomes identical to the formula at 

 the beginning of this paragraph, and so we find for the order the 



