( 427 ) 



w = 3 (r s < + 1) — 2 (;• -h .v + t) — («' r + /i' .:j + y' /) — rf (r + s + i). 



F'rom this we see fluit the order of the lociis proper is lowered by r 

 on account of u common basepoint .'1^^ . Tf there are no points 

 Arst {(f = 0) one can easily account for that lowering of order 

 bv noticing that from the total locus the ('. passing through 

 Ast separates itself, as not belonging to the locus proper. The point 

 Ast furnishes namely together with an arbitrary point of that C- a pair of 

 points satisfying the question ; of which points however only the latter is 

 movable ^). Farthermore we see that a point Arst diminishes the order 

 of L by r-\--s-{-t, a fact one cannot account for by separation, the 

 total locus becoming indetinite '). 



3. The locus proper L has in the basepoints of the three pencils 

 multiple points, the multiplicities of which are easy to determine. 



A basepoint A,- of the pencil (Cj) only is an (6-^ — « .— l)-fold 

 point of L. In fact, the curves Cs and Q passing through Ar have, 

 A,, and the basepoints excepted, still dt — a^l points of intersection 

 each of which combined with .1,. furnishes a pair of points satisfying 

 the question. The tangents in .1,. to the curves CV passing through 

 the st — a — 1 mentioned points of intersection are the tangents of 

 L in the multiple point. 



To determine the multiplicity of a point Agt we remark that to 

 obtain a pair of points satisfying the question and of which one of the 

 movable points coincides with Ast , it is necessary for C]- to pass 

 through Ast (by which it is determined), whilst Cs and Q ^vhich 

 always pass through Ast niust present a movable point of intersection 

 in Ast , thus must touch each other in Ast . The question now rises : 

 How often do two curves Cs and Ct touching each other in Agt in- 

 tersect each other again on the curve C,- passing through Agt ? To 

 answer this question we introduce an arbitrary Cs intersecting the 

 above mentioned C,- in r.i — y — 1 points differing from the basepoints. 

 Through each of these points we allow a Ci to pass which gives 

 rise to a correspondence between the curves 6s and Ct (so likewise 

 between its tangents in Ast) where rs — y — 1 curves C< correspond 

 to a Cs and rt — (■? — J curves Cs to a Ct. Thus for the curves Cs 

 and Ct touching each other in Ast it happens (/'.v -}- ^'^^ — /? — 7 — 2) 



1) If Asi counts foi- f lixed points of intersection of the curves C, and C/, the 

 Gr passing through Asi separates itself i times by whicli tlie degree of i is lowered 

 by it: 



'^) If Arsi counts for .-- lixed points oi' intersection of Cs and Ci, for t iixed points 

 of intersection of Cr and Ct and for y. iixed points of intersection of 6V and G, then 

 Arti diminishes the order of L by e r -}- C s + y.t; this holds for a point ,1,, too, 

 but then we must n-gard 2^ and y, as being zero. 



28 



Proceedings Royal Acad. Amsterdam. Vol. IX 



