( B22 ) 



passing e times tlirougli oacli of the imaginary circle points at in finity. 



Let the plane curve be of order (x, of class v and let t represent 

 the number of its inflectional points. Then the singularities of the 

 evolute or of the cuspidal curve of its focal developable are the following : 

 rank, r = 2 (ft -f p — 2? — ^) 

 class, m=i 2v 



number of stationary planes, « = 2t 



double osculating planes, G =z v^ — v — ft — 3c -f 3ö* -j- 2f — ö 

 stationary tangents, v = 

 double points, H = 3 (ft — r) -\- i 

 double tangents, to = 

 order, w = 2 (3 ft -f t — Ce — 3(j) 

 stationar}^ points /? = 2 (6ft — 2v -\- 3i — Vie — 6<7) 

 stationai-y points not at infinity and not in the i)lane of the curve 



/3' = 2 (5ft — 3r H- 3t — 8f= — 3ö) 

 order of the nodal curve x = 2 (ft + i')" ~ I'^f* ~ 2" — 3t — 8ftf 



— 4ftö — 8rf — 4r<7 + 8f ••' + 88(7 -f 2ö' 

 + 20g -f lOrr. 



The chief singularities of the focal cur^■e are : 

 order, n = 2fi^ -f 4ftr -\- v"' — 1 1ft — v — 3t — 8ft8 — 4ftö — Srs — 2rij 



-f 8f^ -f 8g(7 + Ö*' +20f + 9o' 

 rank, r = 4fir + r' — 4ft — Av — "eve — Ira — 3ö' -f- 8g + 5(j 

 number of stationary tangents, v = 



class, m = 6fx' + G.ar -f- 4ftt -[- 2rt — 3Gft — 12r — 18t — 24ftg 

 _ 6ftö — 12re — Avö — 8tf — 2t(J + 24g- + 12eo' — 80' 

 4- 606 + 28 Ö 

 number of stationary points j? = 2 (3fx + t) (2ft -f r) — 57ft + 21y 



— 27t — 48fte — 18fi(j — 12r8 — Ava 



— 8t8 — 2tö + 48€^ + 366Ö + 4j' 

 + 36ga + 96e -f 40(T. 



Mathematics. — "(>/i /A^^ pos/tioii of the three pohits which a 

 tiristed curve has in coniinon tntth its osculating lüane.'' By 

 Dr. W. A. Versluys. (Communicated by Prof. P. H. Schoute.) 

 (Communicated in the meeting of January 30, 1904). 



§ 1. Let d be the section of the osculating plane V in a point 

 r of the twisted curve C with the developable of which C is the 

 cuspidal curve; then the twisted cui've C and the section d have in 

 the point P only two points in common, that is they have in 

 P a contact of the first order. 



