( 501 ) 



^. + «3 + • • • + ^6 = O , t,+t\ + ...i- t\, = O 



from which ensues that the total iiilersection lies on the surface 



(«, — 1\) + (^ — i\) + . • . + (^. — «\o) = 0, 

 admitting a threefold point in the origin. So the completing curve 

 d''-'^ must pass once through D, the curves (>!•-* doing this together 



five times; moreover the tangent to ^'"^^ in D lies on the common 

 tangential cone of 0]^ and 0^^ in D. 



~ be 



Besides the twenty common nodes D each of tiie two surfaces 

 admits 100 nodes more, which we will represent by Eb and E^. 

 The 100 points Eb corresponding by 20 to the pair {h,b') and four 

 of the pairs {ai,a'i) lie on the curves ()!° and therefore on ()['' ; 



for the same reason Oj^ contains the 100 points Ec. So the total 



intersection of the surfaces 0[^ and (9'^* passes twice through the 



be' '-' 



200 points Eb and E,, these points being nodes of one of the surfaces 

 and ordinary points of the other; as the five curves (>!° pass together 



once through these points, the completing intersection qY^^ must 



contain the 200 points^). 



5. In order to be able to determine the number of points emitting 

 transversals lying on a quadratic cone to eight pairs of lines crossing 

 each other we still want to know how many points the curve o!'*^ 



has in common with each of the 14 given [ines {m, a'l), {b , h'), {c , c') 

 and with each of the 42 transversals of these seven pairs by two. 

 Evidently the first number is 16; for the surface 0^^ corresponding 

 to six of the seven pairs is cut by each line of the seventh pair in 

 16 points. Moreover by means of the method of art. 2 we find for 

 the second number, represented there in general by p, for n = 2 

 the result 12. 



6. We now pass to the determination of the number of points 

 F, emitting transversals lying on a quadratic cone to eight given 

 pairs {rii , a'i), / = 1 , 2, 3, 4, 5, and (6 , //), (c , c'), [d , d'). To that end 

 we consider the three systems 



1) It is quite natural that the points Eh and Ec lie on pj^^. For if Eb lies on p^" 

 the cone with respect to 0^^ consists of the plane x through the transversals 

 ^'^'3'^'^ ^"^ ^^'^^ plane /3 through the transversals ta^,tc, whilst the cone with 

 respect to 0\^ consists of a, and an arbitrary plane through t„ , for which we 

 can take (3 as well. 



34 



Proceedings Royal Acad. Amsterdam. Vol. XIV. 



