( 56& ) 



occurs five times 1 ). So there remain a -f- b -\- c -\- d — 7 planes V rstu 



which are the tangential planes of L in the point A ra t u , so that 

 B rs tu is a (a -\- h -f- c -j- d — 7) fold curve of L. 



10. So we find : 



Of the locus proper L of the pairs of points I* and P' the 

 base-curve B r of the pencil (F r ) is (a — lyfold curve, the common 

 part B IS of the base-curves B r and B s is {a -j- b — ty-fold curve, 

 the common part B rst of the base-curves B r , B s and B t is {a -j- b -f- c — 5) 

 fold curve and the common part B rMa of the four base-curves is 

 (a -f- b -\- c -j- d — lyfold curve. The points of intersection of the 

 base-curves are conic points of L, namely a point of intersection of 

 B r and B s is {a -J- b — %)-fold point, a point of intersection of 

 B n B s and B t or of B r and B st is {a -f- b -f- c — ty-fold point 

 and a, point of intersection of B r , B s , In and B u or of B r , B s and 

 B ta or of B,s and B tu or of B r and B sta is (a -j- b -f- c -j- d — 6)- 

 fold point. 2 ) 



11. The base-curves of the pencils are not the only singular 

 curves of the surface L. There are namely oo 1 triplets of points 

 lying on a surface of each of the pencils. These triplets of points 

 form a double curve of L. If P,P',P" is such a triplet and if Pi 

 and P2 are the sheets through P of the surface, then the sheets 

 P'\ and P"2 correspond to them. Through P' passes another sheet 

 P'3 and through P" a sheet P"3 which sheets correspond mutually. 

 The pair of points not lying on the base-curves is movable along the 

 sheets PI, P'l, along the sheets PI, P"2 and along the sheets 

 P"6, P"3; on the base-curve a third point then joins the pair. 



Further there is still a finite number of quadruples of points, 



!) The number five is found in the following way. The tangents of the movable 



intersections of surfaces F s and Ft touching each other in A rs iu form a cubic cone 

 having the tangent m, t tu to B r .uu as double line. Such an intersection shows to 

 the surface Fr a contact of order two when it touches the movable intersection 



(4) 



of F r and Ft, so if its tangent in Aratu bes on the cubic cone belonging to the 

 pencils (F r ) and {Ft). As this last cone has also m rs tu as double edge, both cones 

 have 9 — 4 = 5 lines of intersection differing from m r ,i u which connected with 

 Wlrttu furnish the five planes under consideration. 



2 ) If the total locus is not indefinite, i. o. w. if there is no point common to 

 the four base-curves then B r is a (stu — l)fold curve and B ri a (stu-^rtu — 2) 

 fold curve of the total locus whilst a point of intersection of B r and Bs is a 

 {stu + rtu — 2)-fold point and a point of intersection of B r , B» and Bt or of 

 B r and B $l a {stu + rtu + rsu — 3) fold point of it. 



