Mathematics. - On the symbolical method. Il. By E. M. BRUINS. (Com~ municated by Prof. L. E. J. BROUWER.)

(Communicated at the meeting of November 27, 1948.)

§ 5. Six points on aconic.

It is generally accepted that PASCAL discovered his theorem before the year 1644. Not before 1806 BRIANCHON found the dual theorem and after~ wards lead the work of STEINER (1828), KIRKMAN (1849), CAYLEY-SALMON ( 1849) to the discovery of the VERONESE~properties of Hexagramma mysticum by VERONESE (1877 - Atti Linc., 3-1, 649-703 seq.). Not only the long period elapsed during this development but also the fact that prominent mathematicians as STEINER, HESSE, SCHRÖTER made statements and suggestions which were proved to be false, indicates that from a geometrical point of view these VERONEsE-properties do not belong to the simplest ones. CREMONA's discovery of the relation between Hexagramma mysticum and the lines on a cubic surface with a conical point lead to a more-dimensional treatment of the problem.

In the following it will be shown, that, the general method being founded, and ternary geometry being reduced to binary geometry, the VERONESE-properties are in deed the simplest theorems one can write down, the PASCAL-theorem being proved.

I. PASCAL's theorem.

(lp) (k m) (n i) Pik, mn + (m i) (I n) (pk) Pkl,np + (n k) (mp) (i 1) P lm, pi = 0

i, k, l, m, n, p being six parameters of points on a conic.

Proof.

(lp) (km) aian + (lp) (in) ak am + (mi) (In) akap + (mi) (kp) a/an + + (n k) (mp) al ai + (n k) (I i) am ap­

- (i/) (km) ap a n + (mp) (in) ak al- (m 1) (in) Bk ap + (mi) (In) ak ap + + (ki) (mp) al an + (n k) (mp) al ai + (n k) (I i) am ap =

[ (m i) (I n) - (m I) (i n) - (i I) (m n) ] ap Bk = O. In virtue of the relation Pik, mn = -P mn, ik this relation is invariant

under cyclical permutation and revers al of the order ik l m n p. So we have 60 PASCAL-lines pik I m n p.

11. piklmnp = (li) (pk) anam - (in) (pm) aWk = O.

Pro 0 f : Breaking up the ternary brackets into binary cycles we have from

(ikl) (knp)(mnx) + (nkl)(mnp) (ikx) = 0,

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dividing by (ik) (kl) (kn) (np) (mn) the equation stated in the theorem From this it follows immediately:

piklmnp == ppnmlkl == -pklmnpl.

Transforming am in (li) and ak in (In) we ob ta in

plklmnp = (m i) (p k) an a, + (p I) (k m) an al + (m p) (k n) al a,.

giving the coordinates on the fundamental triangle i, I, n etc.

lIl. The three PAscAL~Hnes

plmknlp, plnkplm, plpkmln, are concurrent in the STEINER~point

Sik I, mnp - 6 1kl - 6 mnp = O.

Proof: SIkl.mnp= likl+lkll+llil+lnml+lpnl+lmpl =

- Pln,mk + Pkp,nl + P ,m, pi =

= PIP,nk + Pkm,pl + Pin, mi = = Plm,pk + P kn, mi + P,P, nl·

The first three PASCAL~points are on pipkmln, the second three on plmknlp and the third three on pinkplm.

As S Ikl, mnp = - Smnp, Ikl and because of the cyclical symmetry in ik! and mnp, th ere are 20 STEINER~points.

A triangle formed by [ik] = 0, [lm] = 0, [np] = 0 is called a VERONEsE~triangle (ik, Im, np) if all i, k, I, m, n, p are different. There are 15 VERONEsE-triangles.

IV. The 15 VERONEsE~triangles are in 20 triples perspective with a STEINER~point as centre.

Proo f: We can split up S ikl, mnp in three other ways:

Sikl, mnp = Pin, mk + Pkm, pi + Pip, ni _ .

= P'm,Pi + Pip,nk + Pnk,lm _ .

-PkP,nl+Pln,ml+Plm,Pk= .

which gives US the vertices of the VERONEsE~triangles

(in, mk, pi), (lm , ip, nk) and (kp, In, mi).

Intersecting corresponding si des of:

(l)and(2): Pmk,lp. P'p,kn, Pnl,ml onplplmkn.

(I) and (3): P km, nl. Pip, im, Pin, kp on plnlpkm,

(2) and (3): Pip,nl, Pnk,im, Pml,pk on plmlnkp.

(1)

(2)

(3)

these three PASCAL-Hnes are the axis of perspectivity of the pairs of triangles and concurrent in Slkl,mpn. the conjugate STEINER-point of

Sik I, mnp.

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V. Two conjugate STEINER~points form the JACOBIAN (common harmonical pair) of 6ikl. 6 m np and the points of intersection of the line joining them with the conic.

Proof: S = Sikl. mnp = ~ikl - ~mnp = 0

SI= Sikl. mpn = ~ikl + ~mnp = 0 from which the first part is evident.

(Q'S) (Q'Sd == (Q' 6ikl)2- (.Q' 6mnp)2 = O. as (Q' 6ikl)2 is in~

dependent of the indices i. k. I. from which the second half follows.

VI. The linear relation between the three in S IIn . kmp concurrent PASCAL~lines is

Proof :

plk/mnp + plm/pnk + piplknm == O.

pik/mnp == (li) (pk)unum - (In) (pm)UiUk

pim/pnk == (Ii)(km)unup- (In) (kp)Uium

piplknm == (li) (mp)unUk - (In) (mk)uiup

Adding these three forms we obtain according to the "fundamental identity (pk)um == (mk)up - (mp)uk etc. a sum == O.

VII. Sikl. mnp _ SikP. mnl + Sinl. mkp + Smkl. inp. Pro 0 f: Inserting the brackets {ik} ... etc. all terms cancel.

VIII. (iln) (kmp) Si/n.kmp + (kin) (imp) SkIn. Imp + + (imn) (kip) Simn.klp + (ilp) (kmn) Si/p.kmn = O.

Pro 0 f: Consider the points

Simn.klp and l Si/n.kmp + ft SkIn. Imp'

Calculating the linear~factors for the line plk/mnp and the second point the coefficient of l vanishes. that of ft is. as the 12 terms cancel nearly all

2 (imn) (kpl) - (kmn) (ipl) (kl) (ip) (mn) ,

whereas the linear~factor of the first point is

2 (ipl) (nkm) - (Ink) (pmi) (kl) (ip) (mn) .

Again. with pkilmnp we find the coefficient of ft vanishing. that of l being

2 (imn) (kpl) - (kmn) (ipl) (il) (kp) (mn) ,

whereas the first point gives

2 (Up) (knm) - (iln) (mkp) (il) (kp) (mn) .

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The th ree points Slmn,klp, Slln,kmp. Skln,lmp are therefore collinearthe relation being

[(ilp) (mkn) - (iln) (mkp)] Siln,kmp + [-(ilp) (kmn) - (Ink) (pmi)] Skln,lmp = [- (imn) (kip) - (kmn) (ipl)] Simn,klp

from which according to theorem VII we have VIII. q.e.d.

Rem ark: The four STEINER-points are collinear on a line of STEINER­.PLÜCKER j (ik, lm, np).

T 'he process of forming the linear-factórs from pand S wilt be given more in details in deducting the equations of the STEINER-PLÜcKER-lines.

IX. j(ik.lm. np) == (kl) (mn) (pi) pklmnpi + (im) (lp) (nk)pimlpnk = O.

Proof:

j(ik.lm. np) == À.piklmnp + f-tPimlpnk. where À.: f-t can be ca1culated from the condition that th is line. through Slln.kmp contains the pointSkln,lmp also.

Now

Skin. Imp = I kil + Iln I + I nkl + I mil + Ipm I + lipl piklmnp = (1'1 (Pk) an a m - (In) (pm) ai ak.

The linear-factor of th is point and this line is given by the sum of twelve linear-factors. However eight cancel and we are left with

(lilk~k) [enk) (mI) + (n!) (mk)] + (lili~k) [(ni) (mp) + (np) (mi)] + - (pm) [(il) (kn) + (in) (kl)] - (In) [(ip) (km) + (im) (kp)] =

2 (ti) (Pkl~Qk) (mI) + 2 (li) ePkli~)P) (mi) - 2 (pm) (il) (kn) - 2 (In) (im) (kp) = 2 (1'1 enk) (mk) (pI) + 2 (im) (kp) (ni) (lp) =

(kl) (ip)-

(iP)(k~(mn) [(imo) (kpl) - (kmn) (ipl)],

which gives the coefficient of À. in the equation for À. : f-t obtained by sub­

stituting Skin. Imp in j(ik.lm. np) = O. The coefficient of f-t becomes:

(nk)(!i) (lp) [(kip) (inm) - (ipt) (kmn)].

which proves the equation for the STEINER-PLÜCKER-line given above. F rom this we have

-(ik) (lm) (np) j(ik.lm. np) == < iklmnp > piklmnp + < imlpnk > pimlpnk

which changes sign under k ~ i and is invariant under the interchange of

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ik. lm; ik. np; lm. np. thus showing that th ere are only 15 STEINER­PLüCKER~lines forming a (203 , 154 ) configuration of three perpective triangles with the STEINER~points.

X. The three PASCAL~lines pkmplin. pmpknli, ppkmlnl are concurrent in the KIRKMAN~point

Klklmnp = I ik I + I kil + Ilm I + I mn I + I np I + I pi I . Proof: Kiklmnp = Pim,nk + Pkp,iI + Pln,pm = = Pil, mk + P kn, pi + P mp, In =

= Pin,pk + Pkm,nl + Plp,im.

The first three points lie on pimpknl. the second three on pilpmkn, the third three on pinlpkm. which proves the theorem.

EVidently

Kiklmnp == Kklmnpl == -Kpnmlki,

so there are only 60 KIRKMAN~points. On the PAScAL~lines through Kiklmnp are six other KIRKMAN~points

forming the triangles Kmpllkn. Kklpinm. Klnmipk and Klmpnki. Kpnikml. Kmlkpnl. The sides of these triangles are PASCAL~lines; corresponding sides meet in three KIRKMAN~points on piklmnp. In this way a DESARGUES~COn~ figuration (103 , 103 ) is generated. There are in all six of these decades of KIRKMAN~points and PASCAL~lines.

XI. The linear relation between the PAScAL~lines concurrent in a KIRKMAN~point is

Proof:

pkmplin + ppkmlnl + pmpknli == O.

pmpknll ~ (Im) (kp) an al - (In) (ki) am ap

- plnlmkp = (ik) (pm) an al - (il) (pn) ak am

pimpknl = (pi) (Im) an ak - (pn) (Ik) al am.

Adding we obtain. carrying out the indicated identical transformations

am (ki) [(pn)al + (lp)an - (In)ap] == O. q.e.d.

XII. (iklmnp) Klklmnp = (klmnp) a: + (ikmnp) a~ + (iklmp) a~ = _ (ilmnp) a~ + (iklnp) a~ + (iklmn) a~.

Pro 0 f: Dividing by < ikImnp > and carryin.g out the only possible identical transformations we have

(pk)a: (km) a~ (mp) a~ _. . (Pi) (ik) + (kl) (lm) + (mn) (np) = Iptl + I,kl + I kil + Ilm I + I mn 1+ Inpl

q.e.d. XIII. Kiklmnp + Kimlpnk + Kiplknm == O.

Pro 0 f: Inserting the symbols {ik} ... all terms cancel.

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The corresponding KIRKMAN~points are therefore collinear on a CAYLEY­SALMON~line Cl In, kmp .

Along the same line is evident:

KkmPlin + Kpkmlnl + KmPknll = 2 Skmp,iln

_ _ 2 (ip) (/m) Kkmplln - Kpkmlnl + KmPknll = 2 (Pml, lp + Ppl,m/) = (im) (lp) . Pml,lp.

XIV. (iklmnp) Kiklmnp + <imlpnk) Klmtpnk + (iplknm) KIPlknm = = (inl) (kmp) Slnl,kmp.

Proof:

- 2 (inl) (kmp) Slnl,lcmp = (kmp> l(li)2a~ + (In)2 a~ + (nr)2 a~1 + + (iln) I (km)2 a! + (mp)2 ai + (pk)2 a;' 1 = I (kmp) (In)2 + <kin) (mp)2 +

+ (mln) (pk)2 + <pIn) (mk)2la~ + ... a~ + ... 8~. The left~hand~side is

I (klmnp) + (mlpnk) + (plknm) 1 a~ + ... a~ + ... a~. Now we have

(kmp) (ln)2 + (kin) (mp)2 + <mln) (pk)2 + (pin) (mk)2 = - (kmpin) - (kmpn/) - (kinmp) - <klnpm)

- (plnmk) - (pinkm) - (mlnpk) - (mlnkp) _

- 2 [ < kmpln) + (mpkin) + < mkpln) ] ,

as follows inverting the order in the underlined cycles. On the other hand is

<klmnp) + (mlpnk) + (plknm) = - (/m)2 (knp) - (lp)2 (mnk) - (lk)2 <pnm) - (kminp) - (mplnk) + - (pklnm) = - [(kmpin) + <mpkin) + (mkpln)1

which proves the identity.

The line of CAYLEY-SALMON Ciln,kmp contains the point Slnt,kmp. Starting with the PASCAL~lines through a KIRKMAN~point or through the

STEINER~point on a CAYLEY-SALMON~line we obtain different, equivalent equations.

XV. Ciln,kmp = (kmplin) pkmplln + <mpknli) pmpknll + + <pkmin/) ppkmlnl.

Pro 0 f: The equation is of the form

À pkmptin + f-l ppkminl = 0,

where À : f-l can be obtained by substituting the coordinates of Kimlpnk.

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Now with

ppkminl = (mp) (Ik) an ai - (mn) (li) ap ak

(imlpnk) Kimlpnk = (mlpnk) a~ + (impnk) a~ + (imlpk) a~

we have:

(p'pkminl Kimnplk) -

2 (li) (ki) (pn) (nk) . (imlpnk) [ - (mp) (Ik) (rm) (mp) (nl)- (mn) (mi) (lp) (km) (pi) +

- (mn) (im) (mp) (pi) (kl) + (mn) (im) (mi) (lp) (pk) ] = 2 (li) (ki) ~~;~~:~~m/) (mp) [ _ (np) (Ik) (im) + (mn) (lp) (ki)] =

2 (liknmp) - (limpnk) = 2 (limpkn) - (linkmp) (im) (lp) (nk) (im) (lp) (nk) .

Again

, _ (minlpk) - (milnkp) (Pkmplin KimnPlk) = 2 (im) (lp) (nk) .

So the CAYLEY-SALMON~line is

- [(limpkn) - (linkmp)] pkmplin + [(minlpk) - (milnkp)] ppkminl = = (kmplin) pkmplin + (pkmin/) ppkminl - (mpknli) [Pkmplin + ppkminl]

from which the theorem follows.

EVidently Ci/n, kmp = Clni, mpk = + Ckmp, Ini. Moreover; as is clear from the geometrical point of view

XVI. Ciln,kmp - Ci/n,mpk = o. Proo f: Inserting the pkmpllll, ... and summing up every two terms

containing the same a~product the left~hand~side is

(kmp) (lin) [ (nm) (pk) ai al + (mp) (Ik) an ai + (ip) (km) an al ] + - (kmp) (lin) [(Pi) (ni) ak a m + (kl) (in) a m ap + (mn) (li) ap ak ]

Now: (ip) (km) an al = (ip) (nm) ak al --:- (ip) (nk) a m al = (lp) (nm) alo ai - (li) (nm) ak ap - (ip) (Ik) a m an + (ip) (In) a m ak =

- (Pk) (nm) al ai - (kn (nm) ap ai - (np) (Ik) a m ai + (in) (kl) am ap + + (li) (mn) ak ap + (pi) (nn a m ak = - (pk) (nm) al ai - (kl) (pm) an ai +

+ (in) (kl) a m ap + (li) (mn) ak ap + (pi) (ni) a m ak

so all terms in Ci In, kmp - Ci In, mpk cancel.

There are 20 Cayley~Salmon~lines.

XVII. Ci/n,kmp + Ckln,imp + Cimn,klp + Cilp,kmn = o. Pro 0 f: Inserting the forms all terms cancel because of the identities

Cilll,kmp = Clin,kmp , piklmnp = ppnmlki = - pklmnpi.

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XVIII. Clln,kmp = <iknmlp) piknmlp + <imnplk) pimnplk +

+ <ipnklm) pipnklm. Proof: Substituting Kiklmnp in ;'plknmlp + flPimnplk= 0 we find

for the coefficient of ;. at first

2 (ni) (pk) (mi) (Ik) + 2 (ni) (pk) (mi) (lp) _ 2 (nI) (pm) (kl) (im) + (ik) (pI) (Im) ·

2 (nI) (pm) (kn) (im) (mn)

as all other terms cancel. Multiplying by < iklmnp > this form contains a factor -2 [ (ik) (Im) (np) + (ip) (Ik) (mn)] the remaining factor being

< imnplk > - < imlknp >. Again, the coefficient of fl is found to be

2 (ni) (km) [(I~~~fi) + (l~~n) ] - 2 (ni) (kp) [ (iktk~ml) + (iPln~n) ] '

which multiplied by < iklmnp > contains thc same factor as the coefficient of ;. the remaining factor being

< imlnp > - < iknmlp >. Therefore the form Ciln,kmp is, apart from a constant factor e equal to

the right~hand~side. Specialising I == m, p == k we find

e < ilnklk > pllnklk == < klklin > pklklln

which gives immediately e = 1. q.e.d. As in the case of the STEINER~points a second linear relation between

the Clln,kmp could be derived in an analoguous way. Leaving apart for the moment the question wet her the equation I is irreducible we have directly:

XIX. The CAYLEY-SALMON~lines Ciln, mpk, Cllk, nmp, Cipn, mik, Cmpn, ilk are concurrent in the point

l:(im, nk, lp) = Sinl,kmp + Kiklmnp

Pro 0 f: Splitting up in { }~brackets we have

Sin I, kmp + Kiklmnp = Sik I, nmp + Kinlmkp = Sinp, mik + Kikpmnl = _ Smnl,ipk + Kinpmkl-I in I + 1 nll+llil +1 mkl +lpm I + Ikpl +

+ I ik I + I kil + llm I + I mn I + I np I + I pi I q.e.d.

I (im. nk. lp) is written in four ways as a sum of two points on each of the CAYLEY-SALMON~lines

Evidently the I. c form a (154, 203)~configuration of two perspective tetragons c.q. three perspective tri~sides.

Writing

l:(im,nk,lp) = Pim,kn + P nk, pi + Plp,mi + Pim,nk + Pkn,Pl + Plp,im

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it is evident, that ~ is invariant under the interchange of two elements in a pair, changes sign if two pairs are commuted.

xx. }; (im,lk, np) + ,E (ik , lp, nm) + }; (ip.lm, nk) _ 3 Siln, kmp

Proo f: Splitting up in { }~brackets all terms cancel.

XXI. Klklmnp = P im, nk + Pkn,PI + PIP, im

Pro 0 f: Evident; see XIX. To each KIRKMAN~point corresponds one VERONEsE~triangle. Every

VERONEsE~triangle corresponds to four KIRKMAN~points viz. (im, nk, lp) corresponds to Kiklmnp, Kmklinp, Kinlmkp, Kikpmnl, obtained from one of these by interchanging a pair of opposite indices. If we substitute the coordinates of Kik! m n p in Î.Utan + ftUkam = 0 we obtain as nearly all terms cancel:

(ii) (In) (km) 1 _ (in) (mp) (pk) = O. (kl) (lm) (np) (pi) ft

from which follows, because of the symmetry in I, p :

XXII. Pin,km, Kiklmnp, Kikpmnl are collinear on the VERONESE~line

v (ik. mn) = (in) (Ik) (lm) (pk) (pm) ai an + (km) (Ii) (In) (pi) (pn) ak a m = O.

Evidently: v(ik, mn) :::::: v(ki, nm) :::::: -v(mn, ik) :::::: -v(nm, ki). so there are 90 VERoNEsE~lines, the sides of the 15 tetragons of KIRKMAN~points, corresponding to the 15 VERONESE~triangles.

XXIII. The PASCAL~point Pin, km is incident with two VERoNEsE~lines, which are harmonical to the sides of the hexagon through Pin,km.

Proo f: An interchange of k, m or i, n changes + in - in the equation v(ik, mn) = O. q.e.d.

On the six sides of the K~tetragon are the PASCAL~points:

Pin,km, Pil,mp. Pin, kp , Pik,mn. Pkl,np, Plm,Pi.

The last three points are on piklmnp, whereas the other three are the vertices of a triangle with the sides pikpmnl, pinlmkp, pmklinp. So the K~ tetragon and the corresponding p~tetragram form a DESARGUES (103, 103 )

configuration. In all fifteen of these configurations can be formed from the sixty

PASCAL-lines and KIRKMAN~points, and the 90 VERONESE~lines.

XXIV. v(ik, mn) + v(np,lk) + v(lm, ip) :::::: O.

Proo f: Evident. These three lines are concurrent in a point, the equation of which can

be found by intersecting v (ik, mn) = 0 and v (np, Ik) = 0 which according to the fundamental formulae, dividing by - < kmp > < iln > is

- (imnplk) Pim,PI + (imlknp) Pim,kn + (iknmlp) Pnk,pl = O.

As the left~hand~side is invariant under cyclical permutation and reversal

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of the order iklmnp there are 60 of these points. We denote the left~hand~ side by < iklmnp > ZI klm np and we obtain dividing by < iklmnp >

(im) (PI) . (im) (kn) . - (Im) (pi) [ 1,11 + I mp I] + (ik) (mn) [ I In I + I mk I] +

(kn) (lp) _ + (kl)(np) [I nll + I kp I] =

llil + Ipml + Imll + lipl + linl + Imkl + Inml + Ikil + Inll + + Ikpl + Ilkl + Ipn I = Sinl,kmp - K iklmnp • So we have:

XXV. Ziklmnp = Sinl.kmp - Kiklmnp.

XXVI. The six VERONESE~lines through the vertices of a VERONESE~ triangle are by three concurrent in four points Z.

Proo f: The equations of the lines through the vertices of (îk, Im, np) are

(nI) (nm) (pI) (pm) (ik) ai ak + EI (ni) (nk) (pi) (pk) (Im) "I am = 0 (in) (ip) (kn) (kp) (Im) a, a m + E2 (il) (im) (kl) (km) (np) an a p = 0 E3 (nI) (nm) (pI) (pm) (ik) al Uk + (il) (im) (kl) (km) (np) an ap = O.

where El, E2' ES are -+- 1. The determinant of the coefficients of al ak. al Um. a n ap is

1) (1 ~ Elf2(3)'

where 1) is the discriminant (ik) (il) ... (ip) (kl) ... (l1p).

The four points of intersection are Zlmnklp. Zi/pkmn. Zimpkln. Zilnk~ of which each can be obtained from the others by interchanging opposite indices.

XXVII. The three points Ziklmnp. ZimPlnk. Ziplknm are incident with the CAYLEY-SALMON~line Ci/n, kmp'

Proo f: We have to show that v(ik, nm). v(np, Ik), Ci/n,kmp are con~ current.

Now Ci/n, kmp can be written:

(kmplin) [(mn) (ik) a, Up - (mi) (ip) an ak] + (mpknli) [ (pi) (Im) a n ak - (pn) (Ik) ai a m ] + (pkminl) [ (kl) (np) ai am - (ki) (nm) al a p ] = O.

The determinant of the three equations in ai a m • ak a n • a, ap given by v (ik. nm) = O. v (np, Ik) = 0, C i/n, kmp = 0 is therefore apart from a factor

(Ik) (pn) (Im) (pi) (ik) (mn) . (im) (1n)' (pk) . (kn) (li) (pm) . (pI) (im) (nk) equal to

1 0 o -1 1

[(km) (in) (lp) + (mp) (kn) (Ii)] - [(pk) (nI) (im) + (km) (pI) (in)] - [(mp) (Ii) (nk) + (Pk) (mi) (nI)]

which vanishes.

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XXVIII. Ziklmnp + Zlmlpnk + Zlplknm = 3 Slnl,kmp,

Pro 0 f: The left~hand~side is

3 Slnl, kmp - [ Kiklmnp + Klmlpnk + Klplknm ]

and the form between brackets vanishes.

XXIX. ZkmPlin + Zpkminl + Zmpknli = Slln,kmp.

q.e.d.

XXX. The STEINER~point and the KIRKMAN~points form harmonical groups with the points Zand the SALMoN~points on a CAYLEY-SALMoN~line.

Pro 0 f: We have the relations:

Ziklmnp = Slnl, kmp - Klklmnp

I (im, nk.lp) = Sinl, kmp + Klklmnp •

XXXI. Zkmplln. Zpkmlnl. Zmpknll are collinear on Ziklmnp.

Proo f: We have the relations:

q.e.d.

ZkmPlin = SkiP, mln - Kkmpl/n

Zpkminl = Spnm,kll- Kpkmlnl

l;(im, Ik. np) = SkiP, mln + Kkmplln.

I(tp,lm, nk) = Spnm,kll + Kpkmlnl.

Zmpknll = Smlk,pnl- Kmpknli I (ik, lp. nm) = Smlk,pni + Kmpknll.

Now the STEINER~points are on the line of STEINER-PLÜCKER j(im,lp. nk), the KIRKMAN~points are on piklmnp and the SALMON~points on Ciln,mkp. So the Z~points are collinear on a line through Siln,kmp the fourth harmonical of piklmnp to j(im.lp, nk) and Ciln,mkp.

XXXII. Ziklmnp = 2 (iklmnp) piklmnp + (imlpnk) plmlpnk + + (iplknm) plplknm.

Proof:

Clnl,kmp = (iklmnp) piklmnp + (imlpnk) plmlpnk + (iplknm) piplknm:

- (im) (lp) (nk)j (im.lp. nk) (imlpnk) plmlpnk + (iplknm) plplknm

- Cinl, kmp - (iklmnp) plklmnp.

So Ziklmnp = Clnl, kmp + < iklmnp > plklmnp' q.e.d. We have Ziklmnp + Zimlpnk + Ziplknm = i Cinl,kmp. as is evident from

the relation just deduced.

XXXIII. 1'he linear relation between Zkmplin, Zpkminl, Zmpknli, which are concurrent in Ziklmnp is

Zkmplin + Zpkminl + Zmpknli == O.

Proo f : Zk m plin = Ckip, mln + < kmplin > pk m plin etc. gives by addition for the left~hand~side

Ckip,mln + Cpnm,ki/ + Cmlk,pni + Ci/n,mpk = O.

(To be continued.)