( 29 ) 



Mathematics. — "On pencils of algebraic surfaces." By Prof. 

 Jan dk Vries. 



1. Let ii pciK'il [F") of siii-raccs F" uf order u l)c given, iiiter- 

 secting in tlic base-cur\'e u. 



The ijrineipal laiigciils in a poiiu ,S' of r> lo llie siirfaecs of {F") 

 form a cubic cone having the lajigejit s (o o for edge. 

 For, if {F") is iiulicatecl by 



a" + ;.//; = (1) 



and if i/i- are tlie coonUnates of ,S', (he substitution .r/^. = y/^. -|- (^»c/.. 

 furnishes i]i connection with a'l^ z=z {) and b'y = for a point Z on 

 a principal tangent the conditions 



o?/ <'c +'•''// ^~ = '-' ' "'/ '"^^ ~r^-'*y ''- ^^ "» 

 so that the locus of tiie [)rincipal tangents touching in >S' has as 

 equation 



«-1 ,«-2 2 u-2 n-] ^ J f. .n^ 



If Z is a fixed poiid, )" a variable one, this e(puition represents 

 a surface of order {2n — 3) determining on <J the [)oints >S' which 

 are points of contact of principal tangents thi-ough Z. 



The pt'incljHil tmigenta in points of the base-curve form a con- 

 (jruence of order n^ (2n — 3) and of class 3//,'. 



The inflexional tangents of a pencil of plane curves c" enveloping 

 a curve of class 'Sn (n — 2), the complex of rai/s of the principal 

 taiKjents of i^" is of order 3m {n — 2). 



2. A principal tangent ^j in S becomes four-pointed tangent t^, 



if for a point Z Iving on it the relation a^~ az-\-Xb^i,~ bz^O holds 

 good. So the tangents t^ touching in S belong to the biquadratic 

 cone 



(1 — 1 »i— 3 ,3 ;i-3 jU—l 3 , , 



rty 0,/ a~ b~ — a,i 0,1 a~ o~ =. v . . . . [o 

 As the cones (2) and (3) have the (angeni ^^ represented by 



a;~' «, = , b';~' b,=0 , (4) 



in common, the point aS' will be the point of contact of eleven four- 

 pointed tanyents. 



For (z^ = and /-'I) = the equations (2) and (3) represent the 

 figure formed by the surface of tangejits {s) of o and the scroll r, 

 of the right lines t, having their points of contact on a. 



To determine the order of t^ 1 search for the number of points 



