( 564 ) 



The preceding considerations hold invariably for a point A rs i u 

 lying on the base-curves B r and B s and the common part B, u of 

 the base-curves B t and B u 1 ). 



In a point of intersection A rstu of B rs and B ta the tangential cone 

 is likewise of order a -j- b -f- fi -|- </ — 6 as that cone has the tan- 

 gents m r8 and m tu to 7? /s and B tu as (« -f- 6 — 3) and (c -f- rf — 3) 

 fold edges, whilst in the plane through m r8 and w< u no other right 

 lines Irstu are lying. 



A point of intersection .l,',^ of />, and 7i,,, ( is also &{a-\-b-\-c-\-d — 6) 

 fold point of L as m r and m stu are (rz — 1)- and {b -\- c -f- d — 5)- 

 fold edges of the tangential cone and the only lines of intersection 

 of that cone with the plane through m r and m s(u . 



If finally A rs t u is a point of a common part B r8 tu of the four base- 

 curves, then the point P' of the pair of points VV' coincides with 



Arttu when the surfaces F r , />. Ft and F u have in A r8 tu the same 

 tangential plane V r &tu and all jtass through a same point P. Let us 

 now assume an arbitrary plane V rs t passing through the tangent 



m r stu in ^rsiu to B rstu . The surfaces F r , F 8 and Ft toucliing this plane in 



Arstu cut one another in d — 1 points P, through which we bring 



surfaces F u , of which we call the tangential j)lanes in A rstu V u - 

 Thus we obtain a correspondence, where to V rst correspond d — 1 

 planes 1" (( and reversely to V u correspond <i -\- b -\- c — 1 planes 

 Vrst ; for when V u is given there is between V rs and V t a 

 (c, a -\- b) correspondence, of which V u is plane of coincidence, but 

 not a plane V rs t- ^o there are a -\- b -\- c -\- d ■ — 2 planes of coin- 

 cidence VrstV~ u , of which however Ji ve are not planes V rs tu- These 

 are namely the tangential planes of the surfaces F r , F s and F t of 



which one more point of intersection coincides with A rstu which 



i) It is also easy to sec from the lines of intersection with the plane Va* 

 through the tangents Ms and m t „ to B.< and Bt,, that the tangential cone in 



A? s ]n is of order a + b -f c + d — 6. The line m» counts for b— 1 lines of inter- 

 section, the line m tll for c -\- d — 3. Further, the surfaces F s , Ft and F„ touching 

 Vstn cut one another in a — 2 points not lying on the base-curves; through those 



points we bring surfaces F r , whose tangential planes in Antu cut the plane V>t„ 

 along to lines which lie on the tangential cone. 



