MathematicB. - "Repre.<:enlatiol1 of a Tetralzedral Compie,!: on t!te Points of SjJace," By Prof. JAN 01<: VRIES.

(Communicated at the meeting of April 28, 1923).

1. Let there be givell a pencil of quadratic sut'faces which has a twisted cUl've (14 as base curve. The polal' planes of a point P witll respect to these surfaces pass t hrough a st miglt t I ille lJ, w Itich we shall call the pola?' line of p, Through P there pass two bisecants of ('4; the straight line 71 joins the points of these bisecants which are harmonically separated from P by (14. lf P lies in the vel'tex of olie of tlle fou I' cones belonging to the pencil, the polar lille becomes indetinite; any straight line of the plalle Wk == Ol Om On may he considered in tllis case as apolar line,

The complex of rays T of the polar lines r is represellteel on the space of points I PI. The side Ok 0, is represented in any of the points of lhe opposite side Um 0". lf a stmight line r is to belong to T, its polal' lines 1,1 anel /," with respect 10 tlle 8ul'faces a' and (l' of the pencil, must cut each Ol hel'. If the straight line I'

desel'ihes a plane pencil, 1" and ,." desrribe two projeetive plane pencils; the plane pencil (1') contains accordingly two rays for whieh 1" and 1''' ent each ot hel'. The complex T is therefore quad1'atic 1) and has four cardinal points Ok and fout, cal'dinal planes wk; hence

it is tetrahedral. A point P of (14 is the image of tlle stt'aight line p which touchf\s

Q4 at P. The scroll of the tangents of Q4 is thel'efore represenled in the points of ('4.

2, If P descrihes a sft'aight line 'I', the polar planes of P with respect to a' and {Jo deseribe two projective pencils I'ound the polal' Iines 1" and 1'''. The polar line p describes accol'dingly a quadJ'atic scrolt (p)'; the conjugated scroll consists of the polal' Iines of l'

with I'espect to the quadl'atic surfaces thl'Ough (>4. The points of intersection of 'I' wi~h the earelillal plalles WI., are the imag~s of fOllr

I) IC tbe pencil is defined by LUiXi2 = ° and LbiX.' = 0, the polar plan es of 4 4

the point Y have aiY. and biYi for coordinates, The coordinates of P are in tbis case PI2=(asb4 -a.lb3)YsY4 etc. If we put ala8b~1I4+a2a4blbs=CIS,24' T is represented by C12,34P12P34 + C:l3,a PisPa + CSj,24PSl P:i4 = 0.

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rays p, which pass through the ca.rdinal points Ok. T contains evidenl.ly 00' scrolIs (p)'.

If l' is a ray of T, 1" and rOl cut eaeh othe!', so that the projective pel)(~ils of polal' planes produce aquadratic cone which has Ihe point 1"1'" as vertex. Fl'om this follows that the comple:r cones of

T are I'epresented by the J)oint ran.qes (P) Iying on complea: mys.

3 . The l'ays of T whieh lie ill a plallc 'p (and which accordingly ellvelop the comple,r, conie lp'), al'e represented hy the points P of a twisted cur\'e whiell passes thl'ollgh the ('aldinal points Ok' For Ule intel'sectioll of the pl!l.lIes 'f alld Wk is a tangellt of (p' and is l'epl'esented ill Uf.; . As W4 call ollly conlaill the imagel:i poillts U" UI/I' 0,,, the image of the systam of the tangellts of '1" is a twisted cubie (jl cil'cuml:icribed 10 Ihe letl'ahedl'oll Ol U, U. 0,.

4. 'fhe complex T cuts a linem' comple;v A in a cong1'lleiwe (2,2) which has singIllal' points ill U", sillglllul' planes in Wk. Fo .. Ok is the vel'tex of a plane pellcil belongil1g to both cornplexes, hence 10

(2,2). The polar lines p' and pOl of Ihe I'ays of this plane pencil wilh respecl 1.0 ((. and fJ' fOl'ln two pl'ojective plane peneils in Wk

and these pl'OdllCe a COllie cil'clllllscl'ihed 10 Ol Urn 0". The image of the con,qruence (2,2) is thel'efol'e aquadratie sU1face !,!! cil'eum­

scribed to 0) 0. U. U" As A does nol gener8.1Iy cOlltain ally of the sides UkU" i,2' will

/lol generally cOlltaill ally of thel:ie sides either. I) The 00 6 sllrfages n' are the images of 00 6 congTuences (2,2)

contained in T. To these belong 00' lucial (2,2) defined by the 00'

axilll linea?' complexes.

5. The rays of T belonging to two complexes AI and A" fOI'rn a scroll (p)' of tha fOUl'th ol'del'; this scroll belongs of course at the same time to all eomplexes A of the pencil defined by AI and A" hence also 10 both axial complexes of tlIis pencil. Their axes al'e director li/les of (p )' alld moreover douhle direclor lines, for the complex cone of a point Iying on one of these axes, is cul twice oy the ot her aXls.

I) If A is defined by Ld k/Pk/ = 0, f!2 has for equalion 6

Ld c Y Y = 0, 6 k/ mn m n

Inversely lhe sUl'face Lfk/Y ft Y/ = 0 is lhe image of the (2,2), which is defined 6

(kl by the complex L- pmn = O.

6 Ckl

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The scroll (p)4 is repl'esented by the twisted ctt1've (J' which is the intel'seclion of the two surfaces .Q' that are the images of the

congl'uences defilled by AI and A"

lf the axes 1'1 and 1', of two axial complexes cut each other, the congruence (2,2) which these complexes have ill common wilh 1', degellerates into the system of the complex rays p throllgh the point

R _ 1'11', alld thp. complex rays iu the plane Q = 1\1'" JJl connection with this the image sUl'fares i2' de6ned by 1"1', cul eal'h olhel; in the twisted cm've!,l" represelltillg Ihe t'omplex J'ays in (J, alld ill Ihe polal' line l' of R (the image of the complex cOlle of R); e\"idelllly l' is one of Ihe biseeallts of (l",

11' Q' is an arbilrary twisteJ cubie eÎI'oullJscribed 10 UIO,OIU" there pass 00' sul'faces ,2' thl'ough Q" of which ally two have also ill common a bisecallt of (ll; e\'idelltly Ihey represeflt two axial complexes of whieh the axes cul each olhel', so thaI Ihe cOl'l'espondillg (2,2) splits agaill up illtO a compIe:/: cone and a comple:c eonic; the latte I' is l'epl'esellted by QI,

6. A conic (P,' has four poinls lil common wilh tbe slll'face .Q'

belollgillg to all axial complex A; it is aecordingly tbe image of a mtional scr'ol! (JJ)'. Ally ray s of l' Iyillg in Ihe plalle of (P)', conláills two points of (P)'; the image S of s ca1'l'ies therefol'e two l'ayR of (p)'. Hence the curve (8)1, I'epl'eselltillg the rays s, is the

Jouhle cw've of (p)'. If (P)' passes throllgh Op it is the image of a cllbic seroll (p)1

of which the dOllble directol' line pasi>es tI1l'Ougll 01; fOl' the poillts of intersection of (P)' with W I are the images of two rays p throllgh 01'

lt' (P)' passes thl'ollgh Ol alld tlll'ough . 0" it is Ihe image of ft

quadratic scroll (p)'. IlIvel'sely a scroll (p)' has two I'ays in commoll with an axial complex; its image cuts aeeordillgly the cOl'l'espondillg slll'face J!.' oulside Ok in two points. Hellce this image is eithel' a straight line (~ 2) Ol' a conie throllgh !wo cardinal points U,

7. The poillts P of a plane (P repl'esellt the l'ays of a cOllgl'Uence [p]. The polar plalles a and fJ of P with I'espeel to !wo ql1adratic sllrfaces ti' and p' of Ihe givell peueil form t.wo projeelive shea\'es of plalles 1'01llld the poles of lp. Their inlersecliolls with a plalle t/' form two projeelive lields of rays, hence lP cOlllains three rays

p - afl· The ·planes Cl t1lrough a point Q form a pellcil; one plalle of

the (,ol'I'espondillg peneil ((3) passes through Q, hence Q calTies one

ra)' p.

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The .Iie1d of points r PJ is I herefore I he i mage of a cong1'1lenCe (1,3). This consists of the chords of a twisted cubic Ij3 whieh passes

throllglt the points U; for the range of points (P) ill Wk is Ihe

image of Ihe generalrices ]J of aquadratic cOlle which has (h for verlex.

8. If the twistet! cubic (P)3 passes through three ca1'(li1wl [Joints, it is Ihe image of a c'llbic sCl"oll (/J)'. For all arbitJ'ary surface tIJ'

represellting all axial ~olllplex culs (P)I in three mOre points; Oll

the axis of this complex Ibere rest therefore three lilles of Ilte

scroll. Olie pell~iI (4)1) call ue pllssed I Itrollgh (P)' ; fOl' t 11I'ough ally

fOIlI; poillts of (P)' OOi (1)' eall be passed , each of which con/aills

sevell poin ts of (P)'. The cOI'l'espolld i ng COlli ~"exes A f01'1II also a pencil; Ihe axes of both axial complexes belongillg to Ihis pellcil,

eut all rays of the 8('1'011 alld are Ilterefol'e llte directol' lmes of Ihe cuhic sC1'oll (,,)1.

lf (P)I passes through /100 cllnli1ut! /Joiu/s, it is llte image of a sCl'oll of Ihe foul'th onZe/" 111 t.his case olle lP' passes Iltrough (P)': the scroll belollgs 10 the cOllgruellce (2,2) whieh the con'espolldillg

complex A has ill eomlllOIl with 1'; as it is 1'tlliollIl1, it has a

double cubic.

9, A swflwe [PJn is lhe image of a COH.<.J1'uence wilh sheaf deg/'ee n, for its illtersections wilh a ray t of l' are lhe illlages of 1/ rays

tltrollgh the vertex of Ihe complex COlle represellted by t. The ,1ield degl'ee of the eOllgl'llellee is gellerally 3u fol' each point of inler­sectioIl of [PJn with the cubie //,1 representillg' the rays t Iyillg iJl

a plane lP, is t.he image of n ray of Ihe cOllgruellf'e ill lf' If [PJ" passes Sk times through Oh, the field degree is evident Iy 3n-~sk.

4 A twisted curve (P)" is t.he i mage of a sc/'olt of I he order 2n,

fOl' the image surface r PJ' of all axial eomplex cuts (P)" ill 2n points, which lue the images of as man)' rUJs t cutting Ibe axis of the complex,

10, Ir the oase of a pencil of l)lIadratic surfaee8 !'ollsists of tt

clJbic Q' alJd olie of its ehords, tlle poltu' lilles of Ilie poillts of

spa('e fOl'm a ql1adratie ('omplex whiell i~ I'epl'esellted in the same

wa)' as the tetrahedral complex .

We can alwltys l'ep,'eselll tllis pellt'iJ bJ

a (:c,' - ,v I .'1:.) -+- (:J(x.·-.v • .v.) = 0,

The polal' plalles of tlle poillt y I'elative 10 tlle cones I/ = 0 alld

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fJ = 0 hM'e for coordinates y"-2y,, YIO 0 and 0, y.,-2y., y., The pol ar line of y is thel'efol'e represellted by

Hence

4pu' = Pil Pu' This complex has 0. and O. as cal'dinal pOillts, w. and w, as

cal'dinal plan es, The complex rone of :c touches 0.0, at 0 .. V.O. I\t 0., The

polal' line of y lies in the plane 6 if the equation

g. (2y. ','I:.-Y.Y.,'I:.) -t- ~.y,y,,'I:. + ~, y.y,:J;. + 6. (2Y,':J;,-y ,y.,'I:.) = 0

is salistied by all val lies of ,'I:, and ,'1: " From th is follows thaI the complex rap in S are repl'esented by the poinl s of Ihe cubic which is defined by the cones

2S.y,' + ~,y,y, = ~S,Y. 2S.y,' + ~,y,y, = ;.Y.Y.,

(The chol'd O. O. does not belong 10 the image), The congl'llenee (2,2) which Ihe complex has in common with

the axial complex with directrix ax = 0, b:c = 0 , has for image Ihe quadralic sUl'face Ihe equation of which is

(a.b,) y. y. + [4(o.b.) + (a,b,)] y,y , - (a.b.) Y.Y. + (a.b.) Y,Y. + + 2(a.b,) y,' + 2(a.b.) y,' = 0,