( 177 ) 



tangent of Cj find S^ on an arbitrary plane T"", not passing through 

 the centre of projection 7', is a common tangent of the projection j^i 

 of C\ and of the section x of S.^ with T^. Tlie converse is eqnall}- 

 true. Tlie class of p^ and .s- being i\ and m.^ respectively, the nnm- 

 lier of common tangents of />i and s and thus of C\ and So is }\ m^. 



If S^ be a developable D^, a tangent t of ('i touches D^ if a 

 tangent plane of D^ pass through t, and conversely. Let D\ and 

 C'a be the polar reciprocals of C\ and D.^ To a plane of D.^ 

 passing through a tangent t of C\ corresponds a point of C\ on a 

 generator t' of D\, and conversely. The number of these intersections 

 of C\ and D\ • is i\ ;??.,, for the curve C\ is of order m^ and the 

 developable JJ\ of order 7'i. Then, since there are ;\ rn^ planes of 

 D^ passing through tangents of C^, these are i\ ni^ common tangents 

 of (?i and S^. 



We can show in a very simple way, that if the curve C, be an 

 arbitrary algebraic curve of rank )\, and the surface S^ be an 

 arbitrary algebraic surface of class ni^, the number of common 

 tangents is still 7\ w^. For the tangents to S^ form a complex of 

 order m.^, and the tangents of C\ form a ruled surface of order i\. 

 Now according to a theorem due to Halphp^n ^) the number of their 

 common rays is )\ m„, which proves the proposition. 



§ 3. Some theorems concerning the contact of a developable with 

 'an arbitrary surface will be deduced from the theorem proved above. 



Let C\ be a twisted cubic C' and /)' the developable formed by 

 its tangents. Let S.^ be an arbitrary surface of order ii.^, haviiig a 

 cuspidal and a nodal curve respecti\'ely of order i\ and ^.^ and let 

 D* and S.^ have an ordinary contact in d and a stationary contact in 

 X points, wdiilst none of the tangents of C' is an intlexional tangent 

 of S^ and C- does not touch ^S',. The number of common tangents 

 of 6" and S^ is now also for this particular position i\ m.^ or 4:m^. 

 These common tangents of C^ and ^S^ are: 'P' the tangents of C" 

 touching the curve of intersection .v of B* and ;S'.^ and 2"'^ the 

 tangents of C touching ^'.^ in the points (fand-/ where the surfaces 

 D* and S.^ touch. Let every common tangent of C^ and S^ passing 

 through an ordinary point of contact ö count for .i- common tangents 

 and let every common tangent through a stationary point of contact 

 X count y times. 



The number of common tangents of C^ and x a\ ill be 



4 ;» J — .)! <f — // X- 



1) R. Sturm, Linien Geomelrie, I p. 44. 



