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k congruences G. These have in common with the k corresponding 

 plane pencils k lines p„ . . . pk- 



Let us now find the locus of the line g when the k lines p 

 belong to a same complex of the system C. 



Let {A, a) be any plane pencil. Let us take k — 1 lines of this 

 pencil and let us number them 1, . . . i — 1, i -\- 1, . . . k. 



Each of these lines determines with the corresponding congruence 

 G a linear complex, which has in common with the corresponding 

 plane pencil (P, .t) a line p. The k — 1 lines p found in this way 

 determine a complex of the system C. This complex has a line 

 p-, in common with the plane pencil (P,-,:t,). This line determines 

 with Gi a complex having a line ai in common with [A, a). When 

 i varies from 1 to k we have k series of lines a between which exists 

 a (1,1, ...1) correspondence. There are k coincidences. 



The locus of a right line for which the linear com- 

 plexes that it determines with k fixed linear con- 

 gruences meet k fixed plane pencils in k lines of a 

 linear complex of a system of 6 — k terms is a complex 

 of degree k (order and class) to which belong the given 

 k linea r congruence s. 



If k = 6, we have a theorem of Grassmann. 



4. Let us suppose five groups of three lines H lt . . . H t and five 

 nets of lines R 1 , . . . 7? 6 . 



An arbitrary line g determines with H x , . . . H s five linear congru- 

 ences which meet the five corresponding nets in five lines. If these 

 five lines belong to a selfsame linear congruence the line g describes 

 a congruence. 

 Let .i be a plane. Let us consider in this plane five series of lines p x ,...p t . 



Between the lines of these series it is easy to see that there is 

 such a correspondence that to four light lines corresponds a fifth. 



Let us suppose that three right lines are fixed, whilst the fourth 

 describes a pencil. It is then easy to verify that the fifth also de- 

 scribes a pencil. According to an extension of the principle of Zeuthen 

 there are fifteen coincidences. 



The locus of a right line taken in sue h a way t h at t he 

 linear congruences which it determines with five 

 systems of three lines have in common with five nets 

 five lines of a same linear congruence is a congruence 

 of the fifteenth class. 



In the same way we can verity that this congruence is also of 

 order fifteen and that it contains the generatrices of the same kind as 

 the given lines of the five quadratic surfaces determined by these lines. 



18 



Proceedings Royal Acad. Amsterdam. Vol. X. 



