222 



envelop a developable surface K the points of which represent the 

 circles touching C and the points of the cuspidal curve / of K 

 represent the osculation circles of C. There exists a polar relation 

 between the points, tangents, and planes of osculation of k and the 

 planes of osculation, tangents, and points of /. Dut of the characte- 

 ristic numbers of /; and L we find accordingly through dualisation 

 the characteristic numbers of / and K, for instance: 



Order of /: m =^ i -\- 3;i — 6s — 2a 



Order of K: r = 2ft + r — 4g — <j 



Cusps of / : ^ = 5,x — 3p + 3t - 8f — 3<7 



Order of the nodal curve of /i: x = ^ {(2fi -f i' ^ 4g — of — 13^ — v 



— 3t + 24g + lo\. 



From this follows among others: 



To a given pencil there belong r tangent circles of C, but to a 

 concentric pencil only r — (fi — 2s) in finite space (class of the 

 evolute). If we have 3 curves Ci,C\,C^, the surfaces K^K^Ji^, have 

 in all )\7',)\ points in common, of which however there lie 

 4(ft, — 25,) (fi, — 28,) (ftg — 2?,) in W. The rest is the number of 

 circles touching the 3 curves. 



Through a given point there pass m osculating circles of C. The 

 projection of / out of W on B is the evolute; / passes o times 

 through W, hence the order of the evolute is m — g. The evolute 

 has /? cusps (vertices of C) in finite space and moreover /[^ — 2e — 2<T 

 at infinity, arising because ii — 26 — 2(i tangents of / pass through 

 W, lie in lo and have their points of contact outside W. 



Through a point there pass .r circles touching C twice. The locus 

 of the centres of these bi-tangent circles is the projection of the 

 nodal curve of K. This curve however passes s = {(i — 2s) (ji—2s — 1) — a 

 times through W, so that the order of the projection is only .v — s. 



The number of tangents to / cutting / oiu'e more, is y = rm -[- 

 _j_ j2/-— 14?>i — 6/i. Of these 2ö{(a — 2s — 2) lie in lo through W. 

 The i-est gives the number of circles of curvature touching C once 

 more. 



The number of triple points of A' is: 



t = i \r' — 3r (r -\- n -\- 'Sm) — 58r + 42?i + 7Sm\. 



Of these however 



(.-2.)(,-2.-l)(M-2e-2)_ 



1.2. 3 ^' ^ 



lie in W. The rest gives the number of circles touching C thrice. 



If we work out these foi-mulas they get the same form as those 

 of Prof. DK Vkii« as is to be expected. 



