( 559 ) 

 2 (s 4- * -f ii - 2). 



5. To return to the question which gave rise to the preceding 

 considerations we find for the number of points Q r8 t u on the arbitrary 

 line /, which are the points of intersection of / with the locus 

 proper L : 



ar + bs 4- ct -f du — 2r — 2 (a + * + m — 2) = 



= ar -f bs 4- e< -f du — 2 (/• 4- s f « 4- i*) 4- 4. 



So we find : 



The locus I. of the pairs consisting of Into movable points common 



•to a surface out of each of the pencils (F r ), (F 8 ), {Ft) and (F H ) 

 of orders r, s, t and u, and not lying on the base-curves, is a 

 surface of order 



ar 4- bs 4- ct -\- du — 2 (r 4- s -]- t -\- u) -\- 4. 



Here a is the number of points of intersection not necessarily 

 situated on the base-curves of the pencils (F s ), {F t ) and {F u ); b I In- 

 analogous number for the pencils (F r ), (F t ) and (F u ), etc. 



6. It the pencils have an arbitrary situation with respect to each 

 other, then a = stu, etc., so that then the order of the locus becomes 



4 (rs tu 4- 1) — 2 (r 4- s 4- * 4- u). 



That order is lowered when three of the base-curves have a common 

 point or two of the base-curves have a common part, which 

 lowering of the order can be explained by separation as long as 

 the total locus is detinite, i. e. as long as the four base-curves have 

 no common point and no triplet of base-curves have a common part. 

 For, if Arstu is a common point of four base-curves then the surfaces 

 of the four pencils passing through an entirely arbitrary point P 

 have another second point in common, namely A rs tu', if B stu is a 

 curve forming part of the base-curves B 8 , B t and B u of the pencils 

 {F 8 ), (F t ) and (F u ), then the surfaces of the pencils passing through 

 an arbitrary point P have moreover the points of intersection in 

 common of B 8tu with the surface F, through P; so in both cases 

 the arbitrary point P belongs to the totals locus. 



If the basecurves B 8 , B t and B u have a common point A sttl then 

 on account of that point the number a is diminished by unity without 

 having any influence on b, c and d. The order of L is thus lowered 

 by r on account of it, which is immediately explained by the fact 

 that the surface F r passing through A stu separates itself from the 

 locus. 



