( 560 ) 



It' the base-curves B t and B u have a curve B lu in common of 

 which for convenience we suppose that it does not intersect the 

 base-curves 5, and B s , this B tli lias no influence on c and c?, whilst 

 a is lowered with sm and /> with rm, where m represents the ord^r 

 of the curve B tu ; for, when /%. /'', and F u are three arbitrary 

 surfaces always .->•/// points of intersection lie on B tu . The order of 

 L is thus lowered with 2rsm by B/ u . This can be explained by 

 the fact, that the locus of the curves of intersection C rs of surfaces 

 F, tiiid F s passing through a selfsame point of P>,„ l ) separates itself 

 from the locus of P and P'. That the locus of those curves of inter- 

 section is really <>l' order 2rsm is easily evident from the points of 

 intersection with an arbitrary line I. We can bring through an 

 arbitrary point Q r of / an F, cutting B tu in rm points; through 

 each of those points of intersection we bring an F s , which rm sur- 

 faces F s cul the right line / in rsm points Q s . To Q r correspond 

 rsm points Q s and reversely. The 2rsm coincidences are the points 

 of intersection of / with the locus of the curves of intersection C rs . 



7. The base-curves B r , B s , Bt and />,< of the pencils are morefold 

 curves of the surface L. It' A, is a point of B r bal not of the 

 other base-curves, then J, is an ((< — l)-fold point of L. For, the 

 surfaces F S} Ft and F u through A, intersect one another in a — 1 

 points, not lying on the base-curves, each of which points furnishes 

 together with A, a pair of points satisfying the question. Each point 

 of B r is thus an (a--l)-fold point, i. o. w. B r is (a-l)-fold curve 

 of tin' surface L. 



Let A rs ' ,0 a point of intersection of the base-curves B, and />',, 

 but not a point of B t and B u > An arbitrary point P of the curve of 

 intersection Cm of the surfaces F t and F„ through A rs furnishes now 

 together with A rs a pair of points PI" satisfying the question pro- 

 perly, as A,,, is for each triplet of pencils a movable point of inter- 

 section not lying on the base-curves. If we let P describe the curve 

 Cm, then the tangent l rs in A rs to the curve of intersection of the 

 surfaces F r and F s through P describes the cone of contact of /. in 

 the conic point A rs . The tangents m r and m s in J, s to B, and B s 

 are {a — 1)- resp. {b — l)-fold edges of the cone. This cone is cut 

 by the plane through m r and m s only according to the line mr 

 counting (a — l)-times and the line m s counting (b— l)-times, as 

 another line l rs lying in this plane would determine two surfaces 



M If Btu cuts the curve B s in a point Astu, then the surface F r passing through 

 Asia separates itself from the locus of the curves of intersection Crs . 



