American Mathematical Monthly, volume 117, number 4, April 2010

99 Pages·2010·1.63 MB·English

by  Daniel J. Velleman

M
THE AMERICAN MATHEMATICAL
ONTHLY
®
Volume117,Number4 April2010
Adrian Rice “To a Factor pre`s”: Cayley’s Partial 291
Anticipation of the Weierstrass ℘-Function
Geoffrey Pearce Transitive Decompositions of Graphs and Their 303
Links with Geometry and Origami
Warren P. Johnson Trigonometric Identities a` la Hermite 311
David Burton Quasi-Cauchy Sequences 328
John Coleman
Grahame Bennett p-Free (cid:3)pInequalities 334
NOTES
Timothy Marshall A Short Proof of ζ(2) = π2/6 352
So Eun Park The Group of Symmetries of the Tower of 353
Hanoi Graph
Li Zhou Recurrent Proofs of the Irrationality of Certain 360
Lubomir Markov Trigonometric Values
M. John Curran Groups of Cube-Free Odd Order 363
Mark Kozek An Asymptotic Formula for Goldbach’s Conjecture 365
with Monic Polynomials in Z[x]
PROBLEMS AND 370
SOLUTIONS
REVIEWS
Sanford L. Segal Plato’s Ghost: The Modernist Transformation of 378
Mathematics. By Jeremy Gray
ANOFFICIALPUBLICATIONOFTHEMATHEMATICALASSOCIATIONOFAMERICA
New from the MAA
Voltaire’s Riddle:
Micromégas and the Measure of All Things
Andrew Simoson
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the legend of Isaac Newton discovering gravity
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wrote Micromégas(cid:15)(cid:3)(cid:68)(cid:3)(cid:86)(cid:87)(cid:82)(cid:85)(cid:92)(cid:3)(cid:68)(cid:69)(cid:82)(cid:88)(cid:87)(cid:3)(cid:68)(cid:3)(cid:41)(cid:85)(cid:72)(cid:81)(cid:70)(cid:75)(cid:3)(cid:72)(cid:91)(cid:83)(cid:72)(cid:71)(cid:76)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3)
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measurements of the very small to the very large,
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(cid:69)(cid:79)(cid:68)(cid:70)(cid:78)(cid:3)(cid:75)(cid:82)(cid:79)(cid:72)(cid:3)(cid:71)(cid:85)(cid:82)(cid:83)(cid:83)(cid:72)(cid:71)(cid:3)(cid:68)(cid:87)(cid:3)(cid:87)(cid:75)(cid:72)(cid:3)(cid:72)(cid:68)(cid:85)(cid:87)(cid:75)(cid:183)(cid:86)(cid:3)(cid:72)(cid:84)(cid:88)(cid:68)(cid:87)(cid:82)(cid:85)(cid:3)(cid:80)(cid:88)(cid:86)(cid:87)(cid:3)(cid:69)(cid:72)(cid:3)(cid:68)(cid:3)(cid:75)(cid:92)(cid:83)(cid:82)(cid:70)(cid:92)(cid:70)(cid:79)(cid:82)(cid:76)(cid:71)(cid:15)(cid:3)(cid:90)(cid:75)(cid:92)(cid:3)(cid:68)(cid:81)(cid:3)
(cid:82)(cid:79)(cid:71)(cid:3)(cid:83)(cid:85)(cid:82)(cid:69)(cid:79)(cid:72)(cid:80)(cid:3)(cid:86)(cid:87)(cid:88)(cid:71)(cid:76)(cid:72)(cid:71)(cid:3)(cid:69)(cid:92)(cid:3)(cid:48)(cid:68)(cid:88)(cid:83)(cid:72)(cid:85)(cid:87)(cid:88)(cid:76)(cid:86)(cid:179)(cid:87)(cid:75)(cid:72)(cid:3)(cid:79)(cid:72)(cid:68)(cid:71)(cid:72)(cid:85)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:75)(cid:72)(cid:3)(cid:41)(cid:85)(cid:72)(cid:81)(cid:70)(cid:75)(cid:3)(cid:72)(cid:91)(cid:83)(cid:72)(cid:71)(cid:76)(cid:87)(cid:76)(cid:82)(cid:81)(cid:179)
(cid:76)(cid:86)(cid:3)(cid:68)(cid:3)(cid:83)(cid:88)(cid:85)(cid:86)(cid:88)(cid:76)(cid:87)(cid:3)(cid:70)(cid:88)(cid:85)(cid:89)(cid:72)(cid:15)(cid:3)(cid:68)(cid:81)(cid:71)(cid:3)(cid:90)(cid:75)(cid:92)(cid:3)(cid:76)(cid:81)(cid:3)(cid:80)(cid:72)(cid:68)(cid:86)(cid:88)(cid:85)(cid:76)(cid:81)(cid:74)(cid:3)(cid:83)(cid:75)(cid:72)(cid:81)(cid:82)(cid:80)(cid:72)(cid:81)(cid:68)(cid:3)(cid:90)(cid:72)(cid:3)(cid:86)(cid:82)(cid:80)(cid:72)(cid:87)(cid:76)(cid:80)(cid:72)(cid:86)(cid:3)(cid:74)(cid:72)(cid:87)(cid:3)(cid:76)(cid:87)(cid:3)(cid:3)
(cid:3)(cid:3)(cid:3)(cid:3)(cid:90)(cid:85)(cid:82)(cid:81)(cid:74)(cid:17)(cid:3)(cid:36)(cid:79)(cid:79)(cid:3)(cid:76)(cid:81)(cid:3)(cid:68)(cid:79)(cid:79)(cid:15)(cid:3)(cid:87)(cid:75)(cid:76)(cid:86)(cid:3)(cid:69)(cid:82)(cid:82)(cid:78)(cid:3)(cid:76)(cid:86)(cid:3)(cid:68)(cid:3)(cid:70)(cid:68)(cid:86)(cid:72)(cid:3)(cid:86)(cid:87)(cid:88)(cid:71)(cid:92)(cid:3)(cid:76)(cid:81)(cid:3)(cid:75)(cid:82)(cid:90)(cid:3)(cid:80)(cid:68)(cid:87)(cid:75)(cid:72)(cid:80)(cid:68)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:3)
(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:68)(cid:81)(cid:71)(cid:3)(cid:86)(cid:70)(cid:76)(cid:72)(cid:81)(cid:87)(cid:76)(cid:192)(cid:70)(cid:3)(cid:78)(cid:81)(cid:82)(cid:90)(cid:79)(cid:72)(cid:71)(cid:74)(cid:72)(cid:3)(cid:69)(cid:72)(cid:70)(cid:82)(cid:80)(cid:72)(cid:86)(cid:3)(cid:70)(cid:82)(cid:80)(cid:80)(cid:82)(cid:81)(cid:3)(cid:78)(cid:81)(cid:82)(cid:90)(cid:79)(cid:72)(cid:71)(cid:74)(cid:72)(cid:17)
Catalog Code: DOL-39
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THE AMERICAN MATHEMATICAL
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Volume117,Number4 April2010
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“To a Factor pre`s”: Cayley’s Partial
Anticipation of the Weierstrass ℘-Function
Adrian Rice
1. INTRODUCTION. Today, Arthur Cayley is best remembered for his contribu-
tionstomodernalgebra,principallyintheareasofmatricesandgrouptheory.Indeed,
therecanbeveryfewmathematicianswhohavenotcomeacrosstheCayley-Hamilton
theorem regarding the characteristic equations of square matrices, or investigated the
propertiesofgroupsbymeansofaCayleytable.ButCayleywasfarfrombeingsolely
an algebraist. His mathematical output included fundamental work in, for example,
graphtheory(inparticular,hisinitiationofthestudyoftrees),algebraicgeometry(in-
cludingtheCayley-Salmontheorem,whichestablishedtheexistenceof27linesona
cubicsurface),andprojectivegeometry(wherehisgeneralizedideasonmetricswere
amajorinfluenceonFelixKlein).Althoughhismanyandvariedpublicationsbrought
him well-deserved fame, it was not his foundational work on matrices and abstract
groups that was considered the most significant in his own lifetime [19, p. xvi]. He
wasmostfamousasoneofthefoundersandchiefpractitionersofinvarianttheory,one
of the key areas of 19th-century mathematics, to which many mathematicians con-
tributed,includingSylvester,Salmon,Hermite,Clebsch,Gordan,and,mostfamously,
Hilbert[25].
In a period of just over half a century, Cayley published over 900 research papers
andarticles.Butheonlywroteonefull-lengthbook[14],andinterestingly,ithadnoth-
ing to dowith the areas for which he is best known. It was actually on the subject of
elliptic functions, another dominant area of mathematics at the time and yet another
of Cayley’s major fields of interest. But this side of Cayley’s mathematics has been
largely overlooked by mathematicians and historians of mathematics alike, a relative
neglect not helped by the fact that elliptic functions themselves are studied far less
today than in their heyday of the 19th century, when they attracted the likes of Abel,
Jacobi,Riemann,andWeierstrass.1 Nevertheless,itwasinthetheoryofellipticfunc-
tionsthatCayleymadeanimportantcontribution,whichdoesnotappeartobewidely
known,namely,thefirstpublishedderivationofafundamentalformulaforellipticin-
tegrals,whichanticipatedafamousidentityofWeierstrassbyseveralyears.Butwhat
is particularly pleasing about Cayley’s proof is that it relies totally on his use of in-
variant theory, a subject that, on the face of it, seems to have little connection to the
theoryofellipticfunctions.Inthispaper,wecompareCayley’sderivationtothestan-
dard formula, due to Weierstrass, and discuss reasons for the obscurity of the former
comparedtotherelativefameofthelatter.
2. BACKGROUND:ARTHURCAYLEYANDKARLWEIERSTRASS. Bornin
Richmond,justoutsideLondon,ArthurCayley(1821–1895) enteredTrinityCollege,
Cambridge, in 1838, graduating as senior wrangler (i.e., the first in his class) four
years later. Even before his graduation, his research career had already begun with
his earliest papers appearing in the Cambridge Mathematical Journal in 1841. After
spending four years as a research fellow at Trinity, he moved to London, where he
worked as a conveyancing lawyer for well over a decade. During this time, however,
doi:10.4169/000298910X480766
1Fordetailedaccountsofthedevelopmentofellipticfunctions,see[21]and[23].
April2010] CAYLEY’SANTICIPATIONOFTHEWEIERSTRASS℘-FUNCTION 291
hecontinuedtoproduceanabundanceofmathematicalresearch,publishingover300
papers in this period, not only in British journals, but also in publications overseas,
suchasLiouville’sandCrelle’sjournals.
Bytheearly1860s,CayleyhadestablishedareputationasoneofBritain’sforemost
mathematicians, both at home and abroad. It was fitting, therefore, when in 1863 he
waselectedtotheSadleirianprofessorshipofpuremathematicsatCambridge,aposi-
tion he helduntil the endof his life. While he was atCambridge, his research output
remained exceptionally high and wide-ranging, covering a huge variety of subjects.
Indeed, as his friend James Joseph Sylvester once remarked: “whether the matter he
takes in hand be great or small, ‘nihil tetigit quod non ornavit’2” [29, p. 605]. And,
whilebestrememberedasapuremathematician,Cayleyalsopublishedwidelyinap-
plied mathematics, particularly theoretical dynamics and physical astronomy. But it
was in the abstract domain of pure mathematics that he felt most at home. His Cam-
bridge colleague James Clerk Maxwell was only partially in jest when he wrote that
Cayley’s “soul, too large for vulgar space, in n dimensions flourished unrestricted”
[16,p.586].
Karl Theodor Wilhelm Weierstrass (1815–1897) was born in the town of Osten-
felde, Westphalia in 1815. Entering the University of Bonn in 1834, he enrolled
to study public finance and administration, but found that his love of mathematics
increasingly distracted him from his studies. In particular, he developed a lifelong
fascination for elliptic functions, a subject that had recently been re-invigorated by
the work of Abel and Jacobi.3 However, his neglect of his university studies led to
hisleavingBonnin1838withouttakingthefinalexamination.Duringthe1839–1840
academic year, he attended lectures on elliptic functions, given by Christoph Guder-
mann in Mu¨nster, and shortly afterwards began his own independent research on the
subject[24,pp.351–357].
Beginning in 1841, he was employed as a high school (or Gymnasium) teacher in
Prussia,teachingahostofschool-levelsubjectsinadditiontohisbelovedmathemat-
ics. Like Cayley, he carried out mathematical research in his spare time, but unlike
Cayley, only a small fraction of his work was published. However, a seminal paper
published in Crelle’s Journal in 1854 [32] proved so exceptional that it led directly
toprofessorialappointments,firstattheIndustryInstitute inBerlinin1856,andsub-
sequently at the University of Berlin in 1864. From this point, Weierstrass devoted
himself to crafting a series of graduate-level courses, in which he created much of
thefundamentaltheorybehindrealandcomplexanalysis,andmanyofthetechniques
whicharenowubiquitousinbothsubjects[24],[30].
His lectures attracted students from all over Europe, and even America, as he de-
veloped a reputation for lecturing on subjects that were unavailable anywhere else
[4, pp. 4–9]. Although he published very little, his major ideas and techniques soon
reachedtheoutsideworldviathepublicationsofthosewhohadattendedhislectures,
suchasAdolfHurwitz,Go¨staMittag-Leffler,andHermannAmandusSchwarz[2].As
a result, by the time of his death in 1897, Weierstrass’s name was synonymous with
analyticalrigorandlogicalprecision.
Insomeways,thelivesofCayleyandWeierstrasscanbeseentoshareanumberof
superficialsimilarities.Theybothlivedatthesametime,hadapproximatelyequivalent
lifespans,andbegantheirmathematicalresearchintheearly1840s.Bothworkedout-
side the university sector for well over a decade before assuming positions at highly
prestigious universities. Finally, both planned and began to oversee the publication
2“Hetouchesnothingthathedoesnotadorn.”
3Foradiscussionoftheearlyhistoryofellipticfunctions,see[27].
292 (cid:2)c THEMATHEMATICALASSOCIATIONOF AMERICA [Monthly117
of their respective collected works ([15], [33]) near the end of their careers, but died
beforetheywerecomplete.
But the two men were vastly different in many fundamental respects: Cayley’s
mathematicswasoftenrapidlywrittenandhastilypublished,whileWeierstrass’swas
more painstakingly produced and logically structured. As has been mentioned, Cay-
ley published a huge quantity of his research, often within weeks of its composition,
whereas Weierstrass’s innate perfectionism “invariably compelled him to base any
analysisonafirmfoundation,startingfromafreshapproachandcontinuallyrevising
andexpanding”[3,p.221],withtheconsequencethatmuchofhisbestworkremained
unpublished for many years. A consequence of these respective characteristics was
thatCayley’sworksometimescontainedmistakes,whichhewasonoccasionobliged
to correct in subsequent works, although apparently “the prospect of being in error
did not seem to worry him unduly—after all, a mistake could easily be rectified in a
subsequent corrective ‘Note’” [19, p. 64]. Weierstrass, on the other hand, was highly
criticalandrigorous,withtheresultthathismathematicstendedtobemorethoroughly
thoughtoutandpolished.
Theircontrastingstyleswerealsoevidentintheirlecturerooms.Cayley,ontheone
hand,doesnotappeartohavebeenaparticularly effective lecturer,andrarelyhadan
audience of more than a handful of students at a time. Even then, his lectures were
oftenallbut impossible tofollow, although he“was likely oblivious thathis listeners
found the material unfamiliar or that they found any difficulty in understanding it”
[19, p. 359]. Consequently, he did little to encourage any except the most motivated
studentstoundertakemathematicalresearch.Thisisperhapsthegreatestcontrastwith
Weierstrass,who,atthezenithofhiscareerinBerlin,wasahighlyeffectiveandpop-
ularinstructor,who“achievedthegreatesteffectivenessthroughhislectures,”4 which
on occasion attracted audiences in excess of two hundred [4, pp. 4–5]. It was via the
content of these lectures that Weierstrass was ultimately able to exercise a huge in-
fluence on the subsequent generation of mathematicians. Indeed, it is from material
deliveredinWeierstrass’slecturesinBerlinthatmuchofthemoderntheoryofelliptic
functionsisderived.
3. ELLIPTIC FUNCTIONS. The standard definition of an elliptic function, due
largely to Weierstrass, is as follows. A meromorphic function f : C → C∪{∞} is
saidtobeellipticiftherearenonzeroω ,ω ∈Csuchthatω /ω ∈/ Rand,forz ∈C,
1 2 1 2
f(z+ω )= f(z), i =1,2.
i
Inotherwords,ameromorphicfunctionisellipticifithastwodistinctperiodsthatare
linearlyindependentoverthereals.Historically,however,theprincipalwayofdefining
anellipticfunctionwasastheinverseofanellipticintegral,
(cid:2) (cid:3) (cid:4) (cid:5)
g(t)= R t, P(t) dt,
where R isarationalfunctionand P(t)isacubicorquarticpolynomialwithnomul-
tiplefactors.Forexample,if At4+ Bt3+Ct2+ Dt + E issuchapolynomial,and
(cid:2)
dt
z = g(t)= √ ,
At4+ Bt3+Ct2+ Dt + E
thenitsinverse,g−1(z),isanellipticfunction.
4“Gro¨ßteWirksamkeiterreichteWeierstraßdurchseineVorlesungen.”
April2010] CAYLEY’SANTICIPATIONOFTHEWEIERSTRASS℘-FUNCTION 293
Althoughmanyexamplesofsuchfunctionsexist,themostfamous(andfundamen-
tal) was introduced by Weierstrass in his Berlin lectures, and has been known ever
sinceastheWeierstrass℘-function:
(cid:7) (cid:8)
(cid:6)∞
1 1 1
℘(z)= + − , (1)
z2 (z−(cid:4))2 (cid:4)2
m,n=−∞
(m,n)(cid:8)=(0,0)
where(cid:4)=2mω +2nω andω /ω ∈/ R.Thisfunctionhastwolinearlyindependent
1 2 1 2
periods,namely2ω and2ω ,andisanalyticeverywhereexceptatpointscongruentto
1 2
0modulo2ω and2ω ,whereithasadoublepole.5 Itsderivative
1 2
(cid:6)
℘(cid:9)(z)=−2 (z−(cid:4))−3 (2)
m,n
isalsoanellipticfunction,withthesameperiods.Subtracting4timesthecubeof(1)
fromthesquareof(2)gives
(cid:6)∗ (cid:6)∗
℘(cid:9)2(z)−4℘3(z)=−60 (cid:4)−4z−2−140 (cid:4)−6+O(z2),
m,n m,n
(cid:9)
∗
where indicates that m and n are never zero simultaneously. The infinite series
(cid:9) m,n (cid:9)
60 ∗ (cid:4)−4 and 140 ∗ (cid:4)−6 turn out to be absolutely convergent, so Weierstrass
m,n m,n
abbreviatedthembytheconstants
(cid:6)∗ (cid:6)∗
g =60 (cid:4)−4, g =140 (cid:4)−6.
2 3
m,n m,n
Since℘(z)= z−2+O(z2),wethereforeget
℘(cid:9)2(z)−4℘3(z)+g ℘(z)+g = O(z2).
2 3
Now, the left-hand side of this equation is doubly periodic, with periods 2ω and
1
2ω , and it is analytic everywhere except at points congruent to 0 modulo 2ω and
2 1
2ω . But since it equals O(z2), it must approach 0 as z → 0, so the singularity at 0
2
is removable and, by periodicity, all the singularities are removable and the function
extendstoaboundedentirefunction.Therefore,byLiouville’stheorem,itmustbethe
constant0function,soweobtain
℘(cid:9)2(z)=4℘3(z)−g ℘(z)−g .
2 3
Lettingt =℘(z)inthisdifferentialequationallowsustodeducethefollowingelliptic
integral:
(cid:2)
dt
z = (cid:4) . (3)
4t3−g t −g
2 3
5Although it is not immediately obvious from its definition that the ℘-function is doubly periodic, the
proofofthisfactisrelativelystraightforwardandmaybefoundinanyintroductorybookonthesubject,e.g.,
[1,pp.160–161]or[31,pp.253–254].
294 (cid:2)c THEMATHEMATICALASSOCIATIONOF AMERICA [Monthly117
ThisintegralexpressioniscentraltotheWeierstrassianapproach,sinceitsinverseis
theWeierstrass℘-function.Furthermore,byvirtueofthefactthateveryellipticfunc-
tion is a rational function of ℘ and its derivative ℘(cid:9),6 it turns out that the Weierstrass
℘-functionisthemostcrucialofallellipticfunctions.ThusWeierstrass’sfundamental
ellipticintegral(3)standsattheveryheartofthetheoryofellipticfunctionsitself.
The above derivation of Weierstrass’s integral form for the inverse of ℘(z) is typ-
ical of the style of analysis we now associate with him: formal, rigorous, with an
overridingemphasisonpowerseriesandissuesofconvergenceandanalyticity.Butit
is utterly different from anything like the kind of mathematics undertaken by Arthur
Cayley, whose approach was far more formulaic and computational. Furthermore,
Weierstrass’s theory of the ℘-function would appear to fit entirely within the realm
of complex analysis, a field that Cayley rarely entered. How then does this topic in-
tersect with Cayley’s work, and in particular, what on earth does it have to do with
invarianttheory?
4. CAYLEY’S INVARIANT THEORY. The idea of invariance essentially perme-
ates the whole of mathematics. For example, by virtue of its double periodicity, any
elliptic function f(z) with linearly independent periods ω and ω is invariant under
1 2
thetransformationsz (cid:11)→ z+ω ,z (cid:11)→ z+ω .Duringthe1840sand1850s,principally
1 2
through the work of Cayley and other British mathematicians such as George Boole,
James Joseph Sylvester, and George Salmon, the study of invariant properties in al-
gebraandgeometrycrystalizedintoamajorsubdiscipline, knownasinvariant theory
(see[17],[18],[25],[34]).
Considerthegeneralbinaryform
(cid:10) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11)
n n n n
φ =a xn +a xn−1y+a xn−2y2+···+a yn
0 0 1 1 2 2 n n
and a linear transformation T : x (cid:11)→ αX +βY, y (cid:11)→ γX +δY, where (cid:10) = αδ −
βγ (cid:8)=0andα,β,γ,δ ∈C.7 SupposeT mapsφ to
(cid:10) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11)
n n n n
φ(cid:9) = A Xn + A Xn−1Y + A Xn−2Y2+···+ A Yn.
0 0 1 1 2 2 n n
We say that a homogeneous polynomial K in the coefficients and variables of φ is a
covariantif,forsomenonnegativeintegerk,
K(A ,A ,...,A ;X,Y)=(cid:10)kK(a ,a ,...,a ;x,y).
0 1 n 0 1 n
Ahomogeneouspolynomial K injustthecoefficientsofφ isaninvariantif,forsome
nonnegativeintegerk,
K(A ,A ,...,A )=(cid:10)kK(a ,a ,...,a ).
0 1 n 0 1 n
CayleywouldoftenwritethatthetwoexpressionsforK were“toafactorpre`s”equal,8
or,inotherwords,identicaluptoapowerofthedeterminantofthelineartransforma-
tion.
6Foraproofofthisstandardresult,see,forexample,[1,p.189]or[31,pp.254–255].
7Inotherwords,thetransformationT canbeexpressedasa2×2invertiblematrixwithcomplexentries,
i.e.,T ∈GL2(C).
8ThephraseoccursinseveralofhistenMemoirsonQuantics,forexample[5,p.246],[8,p.638],[9,
pp.422,423],[10,p.435],[12,p.62],[13,p.291].IamindebtedtoDr.TonyCrillyforthispieceofinforma-
tion.
April2010] CAYLEY’SANTICIPATIONOFTHEWEIERSTRASS℘-FUNCTION 295
Asanexample,takethegeneralbinaryquadratic
φ =ax2+2bxy+cy2
andthelineartransformation
T : x (cid:11)→αX +βY, y (cid:11)→γX +δY
(where(cid:10)=αδ−βγ (cid:8)=0),whichmapsφ to
φ(cid:9) =a(αX +βY)2+2b(αX +βY)(γX +δY)+c(γX +δY)2
=(aα2+2bαγ +cγ2)X2+2(aαβ +b(αδ+βγ)+cγδ)XY
+(aβ2+2bβδ+cδ2)Y2
= AX2+2BXY +CY2.
Now let K be the homogeneous polynomial K(a,b,c) = ac−b2. Then, by our ex-
pressionsfor A, B,andC,
AC − B2 =(αδ−βγ)2(ac−b2)
or
K(A,B,C)=(cid:10)2K(a,b,c).
The polynomial K(A,B,C) = AC − B2 is thus “to a factor pre`s” equal to ac−b2.
Inotherwords, K isaninvariantofthequadraticφ =ax2+2bxy+cy2.
Mattersgetconsiderablymorecomplicatedveryquickly,however,asthedegreeof
thebinaryformincreases.Inthecaseofthebinaryquartic
X =ax4+4bx3y+6cx2y2+4dxy3+ey4,
theinvariantsaremuchhardertofind.In1841,Boolediscoveredthatitsdiscriminant,
D,isaninvariantof X,andthreeyearslaterwasabletoshowthat
J =ace+2bcd −ad2−b2e−c3
isanotherinvariant.When,in1845,Cayleywasabletoprovenotonlythat
I =ae−4bd +3c2
isyetanotherinvariant,butalsothat,inmodernterminology, I and J formaminimum
generating set of invariants of X with respect to the group GL (C), Boole quickly
2
followedthisbyproving(“usingabruteforcecalculation”[26,p.24])that
D = I3−27J2.
This kind of dependence relationship between invariants was later called a syzygy
by Sylvester. Indeed, in addition to the terms “invariant,” “covariant,” and “syzygy,”
Sylvester further enriched the subject by his introduction of a number of terms and
296 (cid:2)c THEMATHEMATICALASSOCIATIONOF AMERICA [Monthly117