VAN DER VRIES. — MULTIPLE POINTS OF TWISTED CURVES. 481 



These points are thus determined by the intersection of the curve U V 

 with the surface S. In the case we are considering, the curve U J^has 

 a ££'-tuple point at (0, 0, 0, 1), i.e. there are kk' branches of the curve 

 passing through this point. This point is also a [(& — 1) (k' — 1)] 

 -tuple point on S. As an ordinary point of the curve does not in gen- 

 eral lie on S, no branch of the curve through the multiple point lies 

 on S. It can also be easily shown that no branch of the curve in 

 general touches S at the point (0, 0, 0, 1). Every branch of the curve 

 thus meets the surface S (k — 1) (k f — 1) times at the point (0, 0, 0, 1), 

 that is U, V, and S intersect kk r (k — 1) (k' — 1) times at the point 

 (0, 0, 0, 1). These points are thus included among the /a v (/a — 1) (v — 1) 

 points common to U, V, and S. We therefore have 



p( fl -l)(v-l) = 2H kk' (k - 1) {k< - 1) 



i. e. h = £ [> v Qjl - 1) (v - 1) - kk' (k - 1) (V - 1)]. 



Therefore : — 



A point of multiplicity k k' on a curve that is the complete intersection 

 of two surfaces containing the point as points of multiplicities k and k', 

 respectively, reduces the number of apparent double points of the curve by 

 \kk' (k — 1) (k> — 1). 



3. If two surfaces of orders /a and v that intersect in a curve of order 

 p having a £&'-tuple point of the above kind have stationary contact at 

 a point, the line drawn to this point from an arbitrary point is a cuspidal 

 edge on the cone that has the curve as base and this arbitrary point as 

 vertex. Similarly, the line drawn to a point of ordinary contact of the 

 two surfaces is a double edge on the cone. If the surfaces have t points 

 of ordinary contact and j3 points of stationary contact, we have as the 

 order of the cone m = jxv, as its number of cuspidal edges, (3 = /?, and 

 as its number of double edges, 



8 = h + t = i O v (jjl — 1) - 1) - kk' (k - 1) (k' - 1)] + t. 



Then, by Art. 327, Salmon's Geometry of Three Dimensions (1882), we 

 have 



r = /* v (ji + v — 2) — 2* — 3)8 + kk' (k — 1) (V — 1), 



» = 3/tv0*+ v — 3) + 3kk' (k — 1) (k l — 1) — 6* — 8/8, 



a = 2 fi v (3 ft + 3 v — 10) + 6 k k> (k - 1) (k> - 1) — 12 1 - 15 ft 



VOL. XXXVIII. 31 



