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Mathematics. — "On the number of common tnngetits of a curve 

 and a surface." By Dr. W. A. Versluys. (Communicated by 

 Prof. D. J. Korteweg). 



§ 1. Let Ci be a plane algebraic curve of class r^ and >S, an 

 algebraic surface of class m^. Ever)- tangent of C\ touching S^ is a 

 tangent of the section s of the surface S^ with the plane V of C^. 

 Conversely, each common tangent of C^ and s is a common tangent 

 of C'l and ^S'^. The curves 6\ and s being of the class ?\ and 7/;^ 

 respectively, they have r^ m.^ common tangents. Hence, S.^ and C^ 

 have ?'i m^ common tangents too. 



Let the plane V of C^ occupy the particular position of touching 

 /S's in ff points of ordinary contact and in x i)oints of stationary con- 

 tact, the class of the section ,s' is now 



Hence, the curves .v and C^ have now 



common tangents. Every tangent of C^ passing through one of the 

 points of contact d and / is a common tangent'of C^ and S.^, without 

 being a common tangent of C'j and .<?. In § 8 will be proved, that, 

 if i>i be the developable formed by the tangents of 6\, each generating 

 line of D^ touching S^ in a point ff counts for two common tangents 

 of Cj and >S'2 and each generator of Dj touching S^ in a point -/ 

 counts for three common tangents of C^ and S^. If C\ be a plane curve, 

 the developable D^ is the plane V counted r^ times. Every ordinaiy 

 contact fl gives thus %\ common tangents of C^ and S^, and every 

 stationary contact / gives 3;\ common tangents of C^ aijd S^. Thus, 

 the total number of common tangents is 



r, (m, - 2 ff - 3 x) + 2 d r, + 3 X r, = r, m,. 



If the plane V of C\ and the surface S^ touch along a line, then 

 every tangent of Cj touches >S'.^ and the number of common tangents 

 becomes iniinite. This case presents itself if S^ be a developable and 

 V one of its tangent planes. Every tangent of C\ touches S^ twice, 

 if S^ be a torus and I^ be one of the planes touching S.^ along a circle. 



If 6\ be a curve in space the number of common tangents of C^ 

 and S^ is still r^m^, where i\ is the rank of 6\. This will be proved 

 first for some special curves and surfaces and afterwards for the 

 general case. 



§ 2. Let S,^ be a cone with vertex T; the projection of a common 



1] Versluys, These Proceedings, May 27, 1905. 



