( 563 ) 



of V rs and Pi, bul nol planes V rs t. So there are a-\-b-\-c — 5 planes 



V rs u which are the tangential planes of' L in the point .1,^, i. o. \v. 

 /•>'/>/ w [a -j- 1> -\- c — by/old curve of surface L. 



9. We then consider a common point A rs tu of the four base-curves. 

 We get a pair of points PP' with point P' coinciding with A rstu when 

 F n F s , F t and F u have in A rstu a common tangent /,..,,„ and all 

 pass once again through a selfsame point P. The cc 1 lines l r8iu form 

 the tangential cone of Z in A rstu . To determine the number of lines 

 l r8(u in an arbitrary plane e through A rsiu we take in this plane an 

 arbitrary line l rst through A rstu and we bring through the d — 1 

 points of intersection (not lying on the base-curves) of the surfaces 

 F r , Fs and F t touching l rst the surfaces F u , whose tangential 

 planes in A rs tu cut the plane f according to lines, which we shall 

 call /„. To l r8 t now correspond d — 1 lines I u and to l u correspond 

 a -\-b -\- c — 2 lines l rsU as there exists between I rs and I t when l u 

 is given a (c, a + ^-correspondence, of which /„ and the line of 

 intersection of f with the plane through the tangents in A„tu to B, 

 and Bs are lines of coincidence, but not lines l t8t . So there are 

 a -\- h -\- c -\- d — 3 lines of coincidence l rst I u of which however three 

 are not lines l rstu . The common tangents in A rstu of the surfaces F r , 

 F s and F t possessing three points of intersection coinciding with 

 A rs t u and where therefore the intersection of two of those surfaces shows 

 a contact of order two to the third, form namely a cubic cone l ) of 

 which the lines of intersection with e are lines of coincidence but 

 not lines I rstu . So in f lie a -f- b -\- c -f- d — 6 lines I rs t u , i. o. w. the 

 tangential cone of L in A rstu is of order a -\- b -f- e -{- d — 6 2 ). 



1 ) This is again evident when taking for (F r ) and {Fs) pencils of planes with 

 coplanar axes B r and B s and for (Ft) a pencil of quadratic surfaces passing through 

 a line containing the point of intersection S of B r and B s . The line of intersection 

 of the planes F,- and F s shows only then a contact of order two to Ft when that 

 line of intersection lies entirely on Ft, so that the cone under consideration becomes 

 again the cone of the generatrices of the quadratic surfaces passing through S. 



2 ) That order can also be found out of the lines of intersection with the plane 

 Vrs through the tangents m,- and m in Arstu to B r and B s . Those lines of inter- 

 section are: the line m r , counting (a — l)-times, the line m, counting (b — 1)- 

 times and c -f- d — 4 other lines. This last amount we find by drawing in plane 

 Vr» an arbitrary line // through Arstu . The surface Ft touching // cuts the surfaces 

 F r and F, touching V„ in d — 1 points (not lying on the base-curves) through 

 which points we bring surfaces F u whose tangential planes in Arstu cut the plane 

 Vn according to lines to be called l„. Between the lines ft and l u we now have 

 a (d — 1, c — ^-correspondence of which the nodal tangents in A rs tu of the inter- 

 section of the surfaces F r and Fs touching V rs are lines of coincidence. The 

 remaining c + d — I lines of coincidence are lines l rs tu . 



38* 



