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Classification of Quadrics  Bromwich
Transcript of Classification of Quadrics  Bromwich

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The Classification of QuadricsAuthor(s): T. J. I'A. BromwichReviewed work(s):Source: Transactions of the American Mathematical Society, Vol. 6, No. 3 (Jul., 1905), pp. 275
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THE CLASSIFICATIONOF QUADRICS*BY
T. J. 1'A. BROMWICHIn a former umber f the T rans a c t onst Mr. J. L. COOLIDGE has givenclassified istofthe possible pecies of quadrics n hyperbolic pace; his methodis based on that used by CLEBSCH for discussing the mutual relations of twoquadrics. It seems desirable,however, hat the classificationhouildbe associated with the algebraic problem f reducing two quaternary uadraticformsto canonicaltypes; the solution f this problem s, of course,based on WEIER
STRASSS theory f invariantfactors.t One obviousadvantageof usinlgnvariant factors s a basis of classification s that they an be determinedwithoutany prelimninaryeduction f the equation to the quadric; indeed, t is generally necessary o find he factors eforewe can begin the reduction. Anotheradvantage maybe fourndn thefactthat the canionical ypes given by WEIERSTRASS are themostcompactpossible.?The first lassificationftwo quadrics by meansofinvariant actois s due toSYLVESTER. Fromthe geometrical oint of view, n elaboratediscussionl ascarriedouit, ndependentlyf WEIERSTRASSS smethods, by PAINVIN; T PAINVINtouchesonithe questionof reality,but only in the simplestcase (whentheinvariant actors re linear). According o MUTH,the application o quadrics
*Presented to the Societyat the St. Louis meeting, eptember17, 1904. Received forpublication March4, 1904.tTransactions of the American Mathematical Society, vol. 4 (1903), p. 161.tE emerntartheiler;ee his paper on bilinear and quadratic forms,Berliner Monatsberichte, 1868, p. 310= GesamimelteWerke,ol. 2, p. 19. For a bibliography, onsult a review ofMUTH'S ElementartheilerLeipzig, 1899), in the Bulletin of the American MathematicalSociety, vol. 7 (1901), p. 308. An elementary ccount of the applicationof thistheory o thereductionof two quadratic formswill be found in a paper by the writer n the QuarterlyJournal of Mathematics, vol. 33 (1901), p. 85.QAs we shall see later,someof COOLIDGW'Sypes contain superfluous oefficients, hich naybe removedby suitable linear substitutions.11 hilosophical Magazine, ser. 4, vol. 1 (1851), pp. 119, 295; this paper is, of course,earlierthan WEIERSTRASS'S. But it is notcomplete, s no methodfordetermining he reducing
substitutions s given,and thusit is not certain hattheclassifications exhaustive. In fact thetwo "singular cases" are omitted; i. e., when the quadrics are cones having (i) a colmmontangentplane,whichtheytouchalong a commongenerator ii) a commonvertex.?fNouvelles Annalesde Mathematiques, ser. 2, vol. 7 (1868), pp. 481, 529; vol.8 (1869), pp. 49, 145, 193. The lastsectionof his paper contains listofthe chiefresultsobtained.275
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276 BROMWICH: THE CLASSIFICATION OF QUADRICS [Julyof WEIERSTRASS'S results and of KRONECKER'S,* on the"singularcase") hasbeenfully arriedout bv KILLING.tWEIERSTRASS'S method f determininghe reducing substitutions oes notshow howto modify hem n case we restrict urselves o real quadratic formsand real linear substitutions. This difficulty as first urmounted yKLEINin his celebratedmemoir n the classificationf linecomplexesf the seconddegreet; KLEIN shows therethat if one of the quadraticformshas a knowntype, nd if we are restricted o real formsand real substitutions),hen thepossible invariantfactorsre restricted ya certainrelation. 1 KLEIN givesfulldetailsofall possiblecases, fthe number fvariables s six,and thesignature of the one known quadratic form s zero (see 1 below,for definitionfsignature).BOCHER ? has carriedouta similar lassificationorthe caseoffive ariables,with quadraticform fsignature hree.In whatfollowswe shall consider the case of fourvariables, he given signature being two ? 2), four ? 3), or three ? 4).
? 1. Preliminary resultson REAL transformationsf quadratic forms.Anyreal quadraticform an be expressedn theshape
a1x2 + 2Xt 2+ + a l 2(wherex1 .,* are real and linearlyndependent)n an infinityf ways;but,however he reduction s effected,he number f positive oefficientsndthenumber f negative oefficientsin a1, a2, **," ) are bothfixed. This isSYLVESTER'S law of inertia Traigheitsgesetz)fquadraticforms.?*KRONECKERpublished numberfpapers n this ase; a listwouldbe too ongfor numerationhere. The earlier aperswillbefoundn the econdhalfof volunmeof his works; thelast paper in which hequestionwas finallyettled) ppeared n the Berliner Sitzungsberichte of1890,1891. Alternative ethods avebeengivenby the present riter, roceedings of the London Mathematical Society, vol.32 (1900), pp. 87, 117,326.tDer Flichenbiischelwceiter rdnung,nauguraldissertation,erlin, 1872; unfortunately hismemoir as not been accessible. It does not appearfromMUTH'S references,hether r noKILLING concernshimselfwiuhthequestionofreality.Inauguraldissertation,onn, 868; reprintedn vol. 23 of heMathematische Annalen,p. 539.11Mathematische Annalen, vol. 23, pp. 561, 562; the theorem has been extended to"HyRMITE'S forms" by A. LOEWY, Crelle's Journal fur die Mathematik, vol. 122(1900),p 69; seealsoapaperbythewriter, roceedings of the London MathematicalSociety. vol. 32 (1901), p. 349.QReihenentwickelungenerPotentialtheorie,eipzig, 1894; the classificationof cyclides) will befoundnKapitel3. It isperhapsf nterestonote hatthe results btainedn ?2 below analsobe usedtoclassifyhe yclic urvesi. e. bicircularquartics), mentionedby BOCHER.lPhilosophical Magazine, ser. 4,vol. 2 (1852), p. 138; Philosophical Transactions of London, 1853, . 407. BoRcHARDT (Crelle's Journal furdie Mathlematik,
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1905] BROMWICH: THE CLASSIFICATION OF QUADRICS 277For our presentpurpose, hedifference etween these two nlumberss themost usefulconstant; this difference e call the signatureof the quadraticformn,fterFROBENIUS.*We are now n a position o stateKLEIN'S theorem given n the dissertationalreadycited):If A is a real quadraticform, whose signature is knownto be mr, hen,uhateverquadraticformB may be (provided that it is real), the characteristic determinantIXA  B has at least M REAL invariantfactors of ODDdegree.In the presentapplication,A is the absolutequadric (of hyperbolicpace)and takesthe form
x2 + x2+ x2 2so that tssignature s 2(i. e. 3  1). Accordingly, hateveruadricB maybe,thedeterminantXA  B I hasat least tworeal invariantfactorsf odd degree,Now the degreeof thisdeterminants 4, so thatthe tworeal invariantfactorsmust have degrees 1, 1 or 3, 1. Hence the only possible combinations findicesfor the nvariantfactorsre given by thesets1111; 211; 31.
It is nowclear that complex rootscan occur only n the first et; further,nthissettheremay be one pair of such roots not more). Thus thefourfundanmentalypes re I It III IV1111 1 11 211 31vol. 53 (1857), p. 275) pointsout that JACOBIwas acquainted withthe result as early as 1847;but apparentlyJACOBI ad no occasion to publish it.The restrictions hat x1,X2 i *..., x4must be real and linearly ndependent are essential; forexample, we have(X+iy)2+ (xiy)2=2x2 22y2, and x2 y2+ (X+y)2+(X y)2 =3x2 +3y .
*Crelle's Journal fur die Mathematik, vol. 114 (1894), p. 187: the paper was alsopublishedin theBerliner Sitzungsberichte ofthat year.t This theoremrequiresmodification n case IA I 0 For instanice,f there re threevariables x, y, z and ifA= x2+ y2,B = 2yz, thesignature fA is 2, but there s onlyone nvariantfactor f I A + ,tB whichhas an odd degree the invariantfactorsreactuallyA, yU2. Ofcoursein KLEIN'S workthecase I A I0 is excluded owing to the nature oftheproblem.The modifiedtheoremmay be stated as follows: Let I 2A + tB I haven1 invariantfactorsfthe ype + yc) v, c being eal (includingero) and y being dd; and let 2 bethenumberf nvariantfactorsf he ype Awhere3is even; th.en n, [ n2 mIn the llustration bove we have n,_ 1, n2 1, and the theorem s verified. For a discussionof the last theorem, ee A. LoEwY, Crelle's Journal fur die Mathematik, vol. 1'22(1900), p. 61; BROMWiCH,Proceedings of the London Mathematical Society, vol.32 (1900), p. 162.
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278 BROMWICH: THE CLASSIFICATION OF QUADRICS [Julywhere,n case II, we have one pair ofconjugatecomplexroots, ndicatedbythebar over the third nd fourth ndices.We can now writedown the standard types f ternis whichoccur in thereduced* equations o the quadrics A, B; lists of these ypeswill be fouindnthepapers alreadycitedand in MUTH's Elemnentartheiler.Let c be the typicalreal root fthedeterminantXA  B I,then he tandardtypes re givenby thefollowing able: 
Invariantfactor X c (x C )2 (X c)3Terms n A i 2xy 2xy+ Z2Terms n B i cx2 2cxyAz x2 c(2xy + Z2) + 2zx
wherex, y, z are all real and do notappearin anyother erms.Some modificationsmust, however,be introduced,n order to retain theregular ypefor the absolute A: and so we put (in the secondand third ases)x , 2 4+ qr,obtaining he table:Invariantfactor (X  c 2 ( C )Terms nA t2 _ i22 72 + z2Terms n B c(V _2)?( )2  (t2 _2+Z2)+2z( )Another hange s niecessary,henwehave a pairofcomplex oots; if these rea + i/3,  i/3, he correspondingeducedtermswillbe given by:
Invariantfactor (X a)2 + B2Terms n A 2 yTerms n B ~2[(a i3)(x + iy)2 + (a? i3) (X iy)2] a(x2 _ y2) + 2fywherex and y are real.We have nowsufficientmaterialto writedownall thetypesof quadricsbyinspection.t
?2. Canonical formsfor thepossible quadrics in hyperbolic pace.TypeL. Indices 1111.
A y2+ yA+ yA 2 IBI ==C2eCC4B c1ys ? c2y +3y22 C3 4Y4where 1l C2, C3, c4 are roots fthe determinantalquation n X, iXA  B I O* It is tobe borne n mindthatthe invariantfactorsan be calculated from he equationtoB,given n any form; indeed theymust, n general,be foundbefore he reduction an be carried ut.t Since IA I= 1, there s here no reason to put down the types corresponding o the identicalvanishingof IXA B I; norto an infinite ootof IVAB I= 0.
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1905] BROMWICH: THE CLASSIFICATION OF QUADRICS 279I(a). cl, c2,C3,4 are all different,represented by the symbol [1111].This case gives the ellipsoid and four hyperboloidsfound by COOLIDGE *(pp. 162, 163, 1. c.).I(b). c,= C2or C3 C4, represented y [(11)11]. The case cl = C2 givesthe spheroids nd hyperboloids f revolution C., p. 168); 03 = cives the surfaces of translation C., ? 4, p. 169).I(c). c1C= C2=03 or c 2C2= C4; represented by [(111)1]. These casesappear as a sphere nd an equ?idistanturface, espectivelyC., p. 170).1(d). cl 0 and C3 C4; represented by [(11)(11)]. This gives theanalogue of Cliford's surface n elliptic pace (C., end of ? 3, p. 169).I(e) . = C2 =C3= C4; represented by [(1111)]. This case is trivial,
geometrically,s the quadric coincideswith the absolute; it is, of course, lgebraicallydistinct.Type IL. Indices 11 11.
A =y2+y2+2 _ 2A=y+ y2+ y3 y4, IBI =_c6c2(a2+P2),B =cy2 + C2yA + a(y2 y2)+ 213y3y4where the rootsof IXA B B = 0 are c1, C2, a + ifB,  i.
II(a). Cl, C2 are unequal; represented y [11 11] . This is the surfacewith maginarycenters ivenby COOLIDGE (C., P. 165). But it maybe welltopoint out that the apparently more general expression given by him is not actuallymoregeneral han his it containsfive oefficients,ne of which an be removedby linear transformations) t; and further, his equation has the disadvantage thatits coefficients cannot be put down directly from the determinant XA  B,before reduction.
II(b). cl =02; represented by [(11) 1]. This is a corresponding urface ofrevolutionC. ?3, subcase 4); here COOLIDGE's equation has the rightnumber of arbitrary coefficients, but again they are not those naturally suggestedby the determinant A  B I; namely, a = i (a3  1), / 1 [b2  (a3 )]i.
* I do notstay orepeat OOLIDGE'S ubclassification, hichdepends n the igns nd relativemagnitudesf he 's; and, nfuture, quotehis paperbythe etterC.tMoreexactly,wo of them an be madeequal,thus: Let p be thatroot f thequadraticp+ p = 4b a3 + a4 , which snumericallyess than1. Writea=I(a a4), Pz4(a3 a,) ( lp2)pl
y3 (X3px4)(1 p2)4 Y44(pX3 X4)( 1p2)';thenwefind x2 _ 2 2 2 2 y2)cd"2x3x4 y3 y4, a X 2bx x +a x Y ) + 2y3 y4;whereaj (a3a4) and P [b2 (as +a4)2].
Ofcourse is real,because fthe onditionb > ( a3+ a4 )2.
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280 BROMWICH: THE CLASSIFICATION OF QIJADRICS [JulyIt might e thought hat we could have cases to correspond o I ( c I (d);but this s notso, on thegroundof ourrestrictiono reality.
Type III. Indices 211.A = y2 + y + y2 y2 IBI = c1c2c3,B=c1y1? +2y+c3(y3y)i (Y3y4) 2therootsof IXA  B = 0 beingX= c, C2, C3, C3.1II(a). 01, 02, C3 all different; epresented y ['211]. This gives theelliptic and hyperbolic araboloids * (C., p. 166).III(b). 02 = 03; represented y [(21)1]. This is thehorocyclic araboloid.(C., p. 167).II1(c). cl= 02=3; represented y [(211)]. This is thehorocyclic urfaceof zero curvatureC., p. 170); in thiscase thearbitraryignmustbe positive,to avoid an ideal surface.III(d). Cl C2; represented y [2(11)]. This gives theparaboloids qfrevolution C., p. 169).
TypeIV. Ind'ices31.A = y2+ y2+ y_y + 2 + Y3Y 4B = c yA+ c2(y' + y3  y') + 2y2(y3  yj)the roots of IXA  BI = 0, being cl, c2, C2, C2.IV (acz 0l C2; represented y [31]. This is a surface o whichCOOLIDGEhas notassigneda name; it is the first n p. 167 (C.). The equation giventhere ontainstwo superfluousonstants 2, a; a2 may be replaced by 1, anda3 by 0, withoutoss of generality.t
*Again I do notstop togive thesubclassificatious. But itmaybe well to pointout that thecoefficients3, a4 in COOLIDGE'S equation re not nvariants; heessentialsrec3 1 (a3 + a4)and the signof a3a4). For if a3 a4=2eq2, wheree= 1, and q is real, we can writeYS3y4= q (X  X4), y3 y4=( X3+ X4)/qandthen 3 y2=XX, while
a3x2 + (a4X2_c3(Y_y ) +e Y3y)Similarly n COOLIDGE's typefor II ( b), theconstant is superfluous, xceptas to signl; inthe typefor II ( c ), a mustbe positiveand maybe put equal to + 1.tTo provethisstatement, all COOLIDGE'S equation B; A beingtheequation to the absolute.ThenBA=( 1l)x ( x3x4)[2a22+ a3(X3 X4)]2= (al)X1+2y2(y3y4)
providedhat a2y27= 2a2X2+ a3(X3X4), y3y4= a2 ( x3  x4). NowsubstitutenA for 2in termsOfY2, Y3P 4; thuswe find a3y a 1A =X2 +y22+ (y Y4) [_ A2 + 34(Y3 Y4) + 1 (X3+X4)]2 2 2Call the last expressionn square brackets y3 + y4 ), then
A X2 +Y2 y2 BA=(a l)x2+2y2(y2Y4).
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1905] BROMWICH: THE CLASSIFICATION OF QUADRICS 281IV (b . cl = C2; represented y [ ( 31 ) ]. This is the third urface onsideredon p. 167 (C.); in the equation theregivenwe maytake r = 2 withoutloss ofgenerality.We havenow exhausted all possible quadric surfaces,n hyperbolic pace.There are thus twelve undamental lasses,represented ythesymbols iven nthe table
aas b i c d
II t117 1 (11)11 __ _ _ _ _ _ _ _III 211 (21)1 (211) 2(11)IV 31 (31)
This classification an be completely arried out by an examinationofXA  B , without ny preliminary eduction. But, for the further ubdivisions, certain mount f reduction s necessary;as it is notpossibleto foretell in I which fthefour ootscl, c2, c3,c is to be associatedwiththenegativeterm n A. More can be donien III; forc3 s necessarilyssociatedwith henegative quare; bithere the signof y3 y4)2 cannotbeforetold. In cases IIand IV we can obtaini omplete nformatiolnromthedeternminant;s maybeseen by consideringhe reducedforms.
?3. Quadrics in elliptic pace.The absolute is here a positive uadraticform nd its signatures 4; hencethe nly ndicesossible re1111,andthe oots1, C2,2, C4 are ll real. Thuswe are restricted o the possibilitiesgiven bytherow numbered in the tableat the end of? 2.Hence every uadric n elliptic space can be reducedto theform
c1y2 + c2y2 + c3y2 + c4y = 0the bsolute eingy2+ y2+ y2 + y2 = 0; the coefficientsC,2 c3, C4 are allreal, beingtherootsof IXA  B = 0.The details of thecases are:I (a). [1111]; surfacesnotofrevolution, uledor notaccording s theproduct c1c c3c4s positive r negative.
I (b). [ (11 ) 11 ]; surfacesofrevolutionc = c2 ),ruled or notaccording sC3 4 is positivernegative.I(c). [(111)1]; spheres not ruled), 1= C2= C3.*This fact followsalso fromtwo theoremsdue to WEIERSTRASS (Berliner Monatsberi ch t e, 1858, p. 207; GesammelteWerke, ol. 1, p. 233; the theorems re givenin ?? 3, 4).
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282 BROMWICH: THE CLASSIFICATION OF QUADRICS [JulyI(d). [(11)(11)] ; CLIFFORDS surfaces, ruled) cl = C2j C3= C4.It may be remarked hatthere re no hyperboloidsn any ofthese cases.
? 4. Quadrics in parabolic (Euclidean)space.The classificationfquadrics n Euclidean space is carriedout n mostof theordinary extbooks n solid geometry,nd detailsneed hardlybe given here;but a fewremarksmaybe added,to indicate hecontrast o thecases consideredin ?? 2, 3.In parabolic space the absolute s degenerate;its equationcan be expressedonly n terms f planecoordinates l,2, U3,U4. If thenA = 0 is the absolute, t will be found that the determinantIA = 0, and that thesignatureofA is 3. Thus, if B = 0 is any quadric (in planeco6rdinates)he determinant IXA + ,tB I has
either i) four inear nvariantfactors+ ,c1, X + kC2,X + lC3, ,tt;or (ii) two inear nvariantfactors+ ,acl,X + ,uc2 nd one 2;
where,n each case, c1, C2, C3 or cl1, 2) are real.From these factswe see that,by applyingWEIERSTRASS'Smethodsin conjunction with a slight modification tso as to include the factors ,t, or p2), thereduced ypes re(i ) { = v2 v2+ v32M~~~~~~~cv + c2v + c3v2 c 2~= cltt2 + c2V22 +3V3 + CV41(ii ) {BA = vc2 v2 + v2f
B=cl' v + C v2 t 2fv3Here thecoefficients4, f cannotbe foundfrom hedeterminalnt A + ,aBl;it is niot suallynecessary o find hemmerely orthepurposeof classification.tBut theymust be foundif we requirethe precise ize of the surfaceB: oneniethodhas beengivenbythepresentwriter lsewhere.Anotherpossibilitymayoccur f IB I  0; it may happenthat
IXA + BI = 0,* Either by the modified ormof KLEIN'S theorem iven above; orbya theorem ue to GUN
DELFINGER which s given in his editionofHESSE'S lecturesongeometry nd also in his lectureson conics (p. 67). See MUTH7s Elementartheiler,. 180.tProceedings of the London Mathematical Society, vol. 32 (1900), p. 158.t To distinguishbetweenruled and nonruled urfaces, he sign ofIB I s sufficient? Proceedings of the Cambridge Philosophical Society, vol. 10 (1901), p. 358 (see?3); see also BRUCKEL'S collectionof GUNDELFINGER'Sformulae n Cr l e's Journ furdie Mathematik, vol. 119 (1898), pp. 210, 313.
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1905] BROMWICH: THE CLASSIFICATION OF QUADRICS 283identically. If so, the reducedforms re(iii) {A v 2 [ v2 + v2,
lB c1v + c v2 + c3v2,B beinga conic n the planeat infinity.But if a quadric given in pointco6rdinatess degenerate he method ustexplainedbreaksdown entirely;forthetransformationif such an equationtoplaneco6rdinatess no longerpossible. To meetthisdifficultyn alternativemethodof reduction s explained in ??1, 2 of the paper just quoted; and asinmpleeneral method f classifications obtained. But theprocessbywhichthe result s obtainedis notverveasy; and it maytherefore e worthwhiletostate theresult in orderto make it accessibleto thosewho do not care to gothrough he detailswhich re unavoidable n finding hereducingsubstitution.Let R_ 0 be the givenquadric,expressed n pointco6rdinates, x2,X21 X 4;S = 0 any sphere, hecoefficientsf S having beenso chosen hat theexpression 8(P XI + P2 X2 + P3 X3+p4)2 representshesquareof thetangentfrom(x1,2, X3, X4) to the phere, here1x1+ P2x2+ p3 3 + p4X4= 0 is theplaneat infinity. Write A, forthe determinantIXR  81 and A1forthesyminetricaldeterminantormed yadding fifthow nd columnP1, p2 P3, pP4 0)to AO. Then the elements f classificationret:(i) thefactorsfA,; say1  Xe1,1  XC2, 1  XC3; where1, C2,3 are allreal and somemay be zero.(ii) the first erm say kXa) in the expansionof ( AO/A,) n descendingpowersofX.When the quadric is transformedo orthogonal artesianco6rdinates sayY,, Y2,y3), theterms f the seconddegreecan be reducedto
cy 2 + c2y2 + c3y2and, as to the remaining erms, hree cases are possible, ccording o the valueof the integer . To put thework n as short a formas possible,we assume(from heresults ftheelementarynvestigations)hatR1, S must aketheforms
1 =clyl + 2y2+ C3Y3+ 2fy3 + C4,S= 1+ Y22 + 2(1y1 + 12Y2 + 13y3) + M
*Moreprecisely, he transformedquationeithervanishes denticallyorfails to give sufficientinformation. It should be observedthat thecorresponding ifficultyoes not arise in hyperbolic or elliptic space; for heabsolutemaybe expressedeither n pointor in planecoordinates.t It is the second of these whichconstitutes hechief implificationntroducedbythismethodofclassification.
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284 BROMWICH: THE CLASSIFICATION OF QUADRICS [Julywhere,n R, eitherfis zero,or bothC3,c are zero. It is theneasyto see that
1112(XCI 1)(XC2 1)(XC3 1),A0= Mf2 (XC1 )(c2 1)(XC 1)(Xc4 M) _ 12 )(XC31)
2 (XC3 1 ) (XC1  1 )(fX 13)2(X1 1 ) (XC2 1)]1 being the determiinantf a certain inear substitution. From thiswe findthatc1, C2,3 are the etters btained n (i) from he factors fA,; also that
X4 Xei Xc21cf X13 )22 ~~~~~3Now,according o (ii), kXa iS thefirst erm f thisexpressionwhenexpandedin powersof1/X; thus,we have three ases:(1) f=0; then Xa =X4; so thatk= c4, a1
(2) f =0, C4= 0; herea

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1905] BROMWICH: THE CLASSIFICATION OF QUADRICS 285It is to be noticedthatBARBARIN'S nomenclature iffers n some respectsfromCOOLIDGE'S, which has been used above; the former ives the genericname paraboloid to class II, and horiquadric o class III. Although there san advantage n giving name to each class, yet it does not appear to methatthenameparaboloid s specially uitedto class II, the quadricsof whichhaveimaginary enters.In conclusion, he classificationf conics n the hyperbolic lane, carriedouton the inesof ??1, 2, is:I(a): [2]; c1y2+c2y _c3y =O.(b): [(ll;C(y2 + y2)_ C3y2 =OS anld1y2 + c(2 Yl)= ?(b): [(1)1]; y2+ Y2 _y ? (thebsolute).II: [111]; c1y2+?a(y2 _y)?2/y2y3 ?
111(a): [21]; C1y2 ?2(Y y+)(y2 y3)2=O(b): [(21)]; c1(y +y2 Y3)?(Y2 Y3)
IV: [3]; c1(y2+yy2 _ ) ?2y1(yy3)=2yIn eachcase the absolute s y2 + y2 _ y= 0; and, accordingo KLEIN'Stheorem,here s at leastone real invariant actor fodddegree.The numberingf classesI, II, III correspondso thatof BARBARIN (1. C.pp. 3957), whoomits lass IV; in fact the methodgivenon p. 33 fordetermining he rotation fails f appliedto theconics V; and theseconicscannotbe reducedto theform ontainedn BARBARIN'S equation 201) on p. 40, whichis thebasis of his subsequentwork.It is ofcoursepossible to givea classificationf conics n theellipticand in
theparabolicplanes,bya suitablemodificationfthe work bove; butthenecessary changes re so obviousthat t is hardlyworthwhileto elnumeratehem.March15, 1905
Trans. Am. Math. Soc. 20