Table Of ContentTHE MATHEMATICAL PAPERS OF
SIR WILLIAM ROWAN HAMILTON
DunsinkObservatorybyW.B.Taylor
Cunningham Memoir No.XVI
THE
M AT H E M A T I C A L P A P E R S
OF
SIR WILLIAM ROWAN HAMILTON
VOL. IV
GEOMETRY, ANALYSIS, ASTRONOMY,
PROBABILITY AND FINITE DIFFERENCES,
MISCELLANEOUS
EDITED FOR THE
ROYAL IRISH ACADEMY
BY
B. K. P. SCAIFE, Ph.D., D.Sc. (Eng.), M.RI.A.
FellowEmeritus,
TrinityCollege,Dublin
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CONTENTS
page
Frontispiece
Preface ix
PART I. MANUSCRIPTS
I. SystemsofRays.PartThird 3
II. TwoLetterstoAugustusdeMorgan(1858) 34
III. LettertoHartonAnharmonicCoordinates(1860) 179
PART II. GEOMETRY
(cid:3)
IV. OnSymbolicalGeometry(1846–49)
[CambridgeandDublinMathematicalJournal,1,45–57,137–154,256–263
(1846);2,47–52,130–133,204–209(1847);3,68–84,220–225(1848);
4,84–89,105–118(1849).] 431
V. OnsomenewapplicationsofQuaternionstoGeometry(1849)
[BritishAssociationReport1849,PartII.,p.1.] 501
VI. OnPolygonsinscribedonaSurfaceoftheSecondOrder(1850)
[BritishAssociationReport1850,PartII.,p.2.] 502
VII. SymbolicalExtensionsofQuaternions; andGeometricalApplicationsof
Quaternions(1855)
[ProceedingsoftheRoyalIrishAcademy6,250,260,311(1858).] 503
VIII. OnaGeneralExpressionby Quaternions for Cones of theThird Order
(1857)
[ProceedingsoftheRoyalIrishAcademy6,506,512(1858).] 504
IX. On a certain harmonic property of the envelope of the chord connect-
ingtwocorrespondingpointsoftheHessianofaCubicCone(1857)
[ProceedingsoftheRoyalIrishAcademy6,524(1858).] 505
X. On some applications of Quaternions to Cones of the Third Degree
(1857)
[BritishAssociationReport1857,PartII.,p.3.] 506
XI. OnAnharmonicCo-ordinates(1860)
[ProceedingsoftheRoyalIrishAcademy7,286–289,329,350–354(1862).] 507
XII. OnGeometricalNetsinSpace(1861)
[ProceedingsoftheRoyalIrishAcademy7,532–582(1862).] 516
XIII. OnGeometricalNetsinSpace(1861)
[BritishAssociationReport1861,PartII.,p.4.] 561
(cid:3)
In the case of papers communicated, the date immediately following the title is the date of
communication.
vi CONTENTS
XIV. Elementary Proof that Eight Perimeters of the Regular inscribed Poly-
gonofTwentySidesexceedTwenty-fiveDiametersoftheCircle(1862)
[PhilosophicalMagazine(4S),23,267–269(1862).] 562
XV. OntheLocusoftheOsculatingCircletoaCurveinSpace(1863)
[ProceedingsoftheRoyalIrishAcademy8,394(1864).] 564
XVI. On the Eight Imaginary Umbilical Generatrices of a Central Surface of
theSecondOrder(1864)
[ProceedingsoftheRoyalIrishAcademy8,471(1864).] 565
XVII. OnRo¨ber’sConstructionoftheHeptagon(1864)
[PhilosophicalMagazine(4S),27,124–132(1864).] 566
XVIII. OnaNewSystemofTwoGeneralEquationsofCurvature(1865)
[ProceedingsoftheRoyalIrishAcademy9,302–305(1867).] 574
PART III. ANALYSIS
XIX. On the Error of a received principle of Analysis, respecting Functions
whichvanishwiththeirVariables(1830)
[TransactionsoftheRoyalIrishAcademy16,63–64(1831).] 581
XX. NoteonaPaperontheErrorofareceivedprincipleofAnalysis(1831)
[TransactionsoftheRoyalIrishAcademy16,129–130(1831).] 583
XXI. OnFluctuatingFunctions(1840)
[ProceedingsoftheRoyalIrishAcademy1,475–477(1841).] 584
XXII. OnFluctuatingFunctions(1840)
[TransactionsoftheRoyalIrishAcademy19,264–321(1843).] 586
XXIII. SupplementaryRemarksonFluctuatingFunctions(1842)
[Proceedings of the Royal Irish Academy 2, 232–238 (1844). (Identical with
pp.317–321ofNo.XXII.)] 631
XXIV. NewDemonstrationofFourier’stheorem(1841)
[ProceedingsoftheRoyalIrishAcademy2,129(1844).] 632
XXV. Oncertain discontinuous IntegralsconnectedwiththeDevelopmentof
the Radical which represents the Reciprocal distance between two
Points(1842)
[PhilosophicalMagazine(3S),20,288–294(1842).] 633
XXVI. On a mode of expressing Fluctuating or Arbitrary Functions by mathe-
maticalformulæ(1842)
[BritishAssociationReport1842,PartII.,p.10.] 640
XXVII. On an Expression for the Numbers of Bernoulli by means of a Definite
Integral, and on some connected Processes of Summation and Integra-
tion(1843)
[PhilosophicalMagazine(3S),13,360–367(1843).] 641
XXVIII. OntheIntegrationsofcertainEquations(1854)
[ProceedingsoftheRoyalIrishAcademy6,62–63(1858).] 647
XXIX. OntheSolutionoftheEquationofLaplace’sFunctions(1855)
[ProceedingsoftheRoyalIrishAcademy6,181–185(1858).] 649
CONTENTS vii
XXX. On the Calculation of Numerical Values of a certain class of Multiple
andDefiniteIntegrals(1857)
[PhilosophicalMagazine(4S),14,375–382(1857).] 653
PART IV. ASTRONOMY
XXXI. IntroductoryLectureonAstronomy(1832)
[DublinUniversityReview1,72–85(1833).] 663
XXXII. TheComet(1832)
[DublinPennyJournal,207,208(December1832);and223,224(January
1833).] 673
XXXIII. Review of Arago’s work, ‘‘The Comet’’, translated by Colonel Charles
Gold,London,1833(1833)
[DublinUniversityReview1,365–372(1833).] 676
XXXIV. On the Day of the Vernal Equinox at the time of the Council of Nice
(1842)
[ProceedingsoftheRoyalIrishAcademy2,249–250(1844).] 681
XXXV. OnapproximatingtothecalculationofEclipses(1844)
[ProceedingsoftheRoyalIrishAcademy2,597(1844).] 682
XXXVI. OntheNewPlanet, Metis(1848)
[ProceedingsoftheRoyalIrishAcademy4,169(1850).] 683
XXXVII. Remarks, chiefly Astronomical, on what is known as the Problem of
Hipparchus(1855)
[Hermathena(Dublin)4,480–506(1883).] 684
PART V. PROBABILITY AND FINITE DIFFERENCES
XXXVIII. OnDifferencesandDifferentialsofFunctionsofZero(1831)
[Transactions of the Royal Irish Academy 17, 235–236 (1830). Translation
in Quetelet’s Correspondance Mathe´matique et Physique 8, 235–237
(1834).] 701
XXXIX. OnaTheoremintheCalculusofDifferences(1843)
[BritishAssociationReport1843,PartII.,pp.2–3.] 703
XL. OntheCalculusofProbabilities(1843)
[ProceedingsoftheRoyalIrishAcademy2,420–422(1844).] 705
XLI. On some investigations connected with the Calculus of Probabilities
(1843)
[BritishAssociationReport1843,PartII.,pp.3–4.] 707
PART VI. MISCELLANEOUS
XLII. Onthedoublerefractionof lightaccordingto theprinciples of Fresnel
(ReviewoftwoscientificmemoirsofJamesMacCullaghB.A.)(1830)
[NationalMagazine1,pp.145–149.Dublin,August1830.] 711
viii CONTENTS
XLIII. Address as Secretary of the Dublin Meeting of the British Association
(1835)
[BritishAssociationReport1835,pp.xli–lvi.] 716
XLIV. On a new theory of Logologues: also On a new theory of varying orbits;
and Explanation of the method of investigation pursued by Mr G. B.
Jerrard for accomplishing the solution of equations of the fifth and of
higherdegrees(1835)
[BritishAssociationReport1835,PartII.,p.7.] 728
XLV. New Application of the Calculus of Principal relations; and Exposition
of Mr. Turner’s theorem respecting theseries of odd numbers, and the
cubesandotherpowersofthenaturalnumbers(1837)
[BritishAssociationReport1837,PartII.,p.1.] 729
XLVI. InauguralAddressasPresidentoftheRoyalIrishAcademy(1838)
[ProceedingsoftheRoyalIrishAcademy1,107–120(1841).] 730
XLVII. Address as President of the Royal Irish Academy on Professor Mac-
Cullagh’s Paper on the Laws of Crystalline Reflexion and Refraction
(1838)
[ProceedingsoftheRoyalIrishAcademy1,212–221(1841).] 739
XLVIII. Notice of a singular appearance of the clouds observed on the 16th of
December,1838(1839)
[ProceedingsoftheRoyalIrishAcademy1,249(1841).] 745
XLIX. Address as President of the Royal Irish Academy on Dr. Apjohn’s
researchesontheSpecificHeatofGases(1839)
[ProceedingsoftheRoyalIrishAcademy1,276–284(1841).] 746
L. Address as President of the Royal Irish Academy on Mr. Petrie’s Paper
ontheHistoryandAntiquitiesofTaraHill(1839)
[ProceedingsoftheRoyalIrishAcademy1,350–354(1841).] 752
LI. OnDr.Robinson’sTableofMeanRefractions(1843)
[ProceedingsoftheRoyalIrishAcademy2,400–401(1844).] 755
LII. Address as President of the Royal Irish Academy on Dr. Kane’s Re-
searchesontheNatureofAmmonia(1839)
[ProceedingsoftheRoyalIrishAcademy2,411–419(1844).] 756
LIII. SirW.RowanHamiltonontheElementaryConceptionsofMathematics
(SevenLetterstoViscountAdare(MarchandApril,1835))
[Hermathena(Dublin)3,469–489(1879).] 762
LIV. Address to Academy of Sir W. R. Hamilton on Retiring from the
Presidency(1846)
[ProceedingsoftheRoyalIrishAcademy3,199–201(1846).] 773
LV. ObituaryNoticeofSirW.R.Hamilton(1865)
[ProceedingsoftheRoyalIrishAcademy9,383(1867).] 775
ListoftheworksofSirWilliamRowanHamiltoninapproximateChronologicalorder 776
Indexforthisvolume 795
Combinedindexforallfourvolumes 801
PREFACE
This fourth volume of the Mathematical Papers of Sir William Rowan Hamilton completes
(cid:3)
the project begun, in 1925 , by the instigators and first Editors: Arthur William Conway
(1875–1950)andJohnLightonSynge(1897–1995).ItcontainsHamilton’spublishedpapers
on geometry, analysis, astronomy, probability and finite differences, and a miscellany of
publications including several addresses. There are also three previously unpublished manu-
scripts, namely: the, unfortunately incomplete, Third Part of the Systems of Rays, the earlier
parts of which were published in Volume I; two letters to Augustus De Morgan, one devoted
todefiniteintegralsandtheothertoathirdorderdifferentialequation;andaverylongletter
andpostscriptaddressedtoAndrewSearleHartonAnharmonicCoordinates.
At the end of this volume will be found a list of Hamilton’s papers in approximate
chronologicalorder.Aswellasanindextothisvolume,thereisalsoacombinedindexforall
fourvolumes.
The frontispiece is a view of Dunsink Observatory, where Hamilton lived and worked from
the time of his appointment, in 1827, as Andrews Professor of Astronomy and Royal
Astronomer of Ireland until his death in 1865. It is a reproduction of an aquatinted etching
publishedin1820byWilliamBenjaminSarsfieldTaylor(1781–1850)andwasintendedtobe
partofhisHistoryoftheUniversityofDubliny.
The Royal Irish Academy acknowledges with gratitude the generous financial assistance
towards the publication of this volume from the National University of Ireland; Trinity
College, Dublin; University College, Dublin; University College, Cork; University College,
Galway;TheQueen’sUniversityofBelfast;TheDublinInstituteforAdvancedStudies.
A warm expression of thanks must be paid to the staffs of the Departments of Manuscripts
and of Early Printed Books in the Library of Trinity College, Dublin, and to the staff of the
RoyalIrishAcademy,fortheirwillingandexpertassistanceatalltimes.
For help in the preparation of this volume especial thanks are due to Dr. Ian Elliott, of
Dunsink Observatory, who suggested the frontispiece and who advised on astronomical
matters, and in Trinity College to Dr. B. P. McArdle, Professor Petros Florides, Miss Hannah
O’Connor, and my brother Professor W. Garrett Scaife. Prof. T. J. Gallagher of University
College,Dublin,andmydaughter,Lucy,gaveinvaluablehelpinproofreading.
The care with which the Cambridge University Press has produced this work is most
gratefullyacknowledged.
B.K.P.S
DepartmentofElectronicandElectricalEngineering
TrinityCollege,Dublin
January2000
(cid:3)
See Selected papers of Arthur William Conway, ed. James McConnell, pp. 8–9, Dublin Institute for
AdvancedStudies,Dublin:1953.
y Thehistorywasnotpublisheduntil1845anddidnotcontainthisparticularetching:W.B.S.Taylor
HistoryoftheUniversityofDublin,T.Cadell,London,andJ.Cumming,Dublin:1845.
The odd spelling of ‘Dunsinok’ for Dunsink is puzzling and has not been explained. Dunsink is the
acceptedEnglishversionoftheGaelicname‘Du´nSinche’(‘FortofSinneach’).
Part I
MANUSCRIPTS
