10/26/2015 Clairaut'stheoremWikipedia,thefreeencyclopedia

https://en.wikipedia.org/wiki/Clairaut%27s_theorem 1/3

Figure1:Anellipsoid

Figure2:Wireframerenderingofanellipsoid(oblatespheroid)

Clairaut'stheoremFromWikipedia,thefreeencyclopedia

NottobeconfusedwithClairaut'srelation.Forthetheoremconcerningsymmetryofsecondderivativesofamapping ,seeSymmetryofsecondderivatives.

Clairaut'stheoremisageneralmathematicallawapplyingtospheroidsofrevolution.Publishedin1743byAlexisClaudeClairautinhisThoriedelafiguredelaterre,tiredesprincipesdel'hydrostatique,[1]whichsynthesizedphysicalandgeodeticevidencethattheEarthisanoblaterotationalellipsoid,[2][3]itwasinitiallyusedtorelatethegravityatanypointontheEarth'ssurfacetothepositionofthatpoint,allowingtheellipticityoftheEarthtobecalculatedfrommeasurementsofgravityatdifferentlatitudes.

Contents

1Formula1.1Somiglianaequation

2Geodesy3References

Formula

Clairaut'sformulafortheaccelerationduetogravitygonthesurfaceofaspheroidatlatitude,was:[4][5]

where isthevalueoftheaccelerationofgravityattheequator,mtheratioofthecentrifugalforcetogravityattheequator,andftheflatteningofameridiansectionoftheearth,definedas:

(wherea=semimajoraxis,b=semiminoraxis).

Clairautderivedtheformulaundertheassumptionthatthebodywascomposedofconcentriccoaxialspheroidallayersofconstantdensity.[6]ThisworkwassubsequentlypursuedbyLaplace,whorelaxedtheinitialassumptionthatsurfacesofequaldensitywerespheroids.[7]Stokesshowedin1849thatthetheoremappliedtoanylawofdensitysolongastheexternalsurfaceisaspheroidofequilibrium.[8][9]Ahistoryofthesubject,andmoredetailedequationsforgcanbefoundinKhan.[10]

Somiglianaequation

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10/26/2015 Clairaut'stheoremWikipedia,thefreeencyclopedia

https://en.wikipedia.org/wiki/Clairaut%27s_theorem 2/3

TheaboveexpressionforghasbeensupplantedbytheSomiglianaequation(afterCarloSomigliana):

where,

isthespheroid'seccentricity,squared

isthedefinedgravityattheequatorandpoles,respectively

(formulaconstant)

ForEarth, =9.7803253359ms2 =9.8321849378ms2k=0.00193185265241e2=0.00669437999013:[11][12]

Geodesy

Seealso:Theoreticalgravity

ThespheroidalshapeoftheEarthistheresultoftheinterplaybetweengravityandcentrifugalforcecausedbytheEarth'srotationaboutitsaxis.[13][14]InhisPrincipia,NewtonproposedtheequilibriumshapeofahomogeneousrotatingEarthwasarotationalellipsoidwithaflatteningfgivenby1/230.[15][16]Asaresultgravityincreasesfromtheequatortothepoles.ByapplyingClairaut'stheorem,Laplacewasabletodeducefrom15gravityvaluesthatf=1/330.Amodernestimateis1/298.25642.[17]SeeFigureoftheEarthformoredetail.

ForadetailedaccountoftheconstructionofthereferenceEarthmodelofgeodesy,seeChatfield.[18]

References1. FromthecatalogueofthescientificbooksinthelibraryoftheRoyalSociety.

(http://books.google.com/books?id=3owAAAAAYAAJ&pg=PA134&lpg=PA134&dq=%22Th%C3%A9orie+de+la+figure+de+la+terre%22&source=web&ots=an0JWH3C8&sig=BMkuXfZEsK3p0tzrZ1Jvfcy7hmw&hl=en&sa=X&oi=book_result&resnum=10&ct=result)

2. WolfgangTorge(2001).Geodesy:AnIntroduction(3rded.).WalterdeGruyter.p.10.ISBN3110170728.3. EdwardJohnRouth(2001).ATreatiseonAnalyticalStaticswithNumerousExamples.Vol.2.Adamant

MediaCorporation.p.154.ISBN1402173202.Areprintoftheoriginalworkpublishedin1908byCambridgeUniversityPress.

4. W.W.RouseBallAShortAccountoftheHistoryofMathematics(4thedition,1908)(http://www.maths.tcd.ie/pub/HistMath/People/Clairaut/RouseBall/RB_Clairaut.html)

5. WalterWilliamRouseBall(1901).Ashortaccountofthehistoryofmathematics(3rded.).Macmillan.p.384.

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10/26/2015 Clairaut'stheoremWikipedia,thefreeencyclopedia

https://en.wikipedia.org/wiki/Clairaut%27s_theorem 3/3

6. Poynting,JohnHenryJosephJohnThompson(1907).ATextbookofPhysics,4thEd.London:CharlesGriffin&Co.pp.2223.

7. IsaacTodhunter.AHistoryoftheMathematicalTheoriesofAttractionandtheFigureoftheEarthfromtheTimeofNewtontothatofLaplace.Vol.2.ElibronClassics.ISBN1402117175.Reprintoftheoriginaleditionof1873publishedbyMacmillanandCo.

8. OsmondFisher(1889).PhysicsoftheEarth'sCrust.MacmillanandCo.p.27.9. JohnHenryPoynting&JosephJohnThomson(1907).ATextbookofPhysics.C.Griffin.p.22.

10. NASAcasefileOntheequilibriumfigureoftheearthbyMohammadA.Khan(1968)(http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19690003446_1969003446.pdf)

11. DepartmentofDefenseWorldGeodeticSystem1984ItsDefinitionandRelationshipswithLocalGeodeticSystems,NIMATR8350.2,3rded.,Tbl.3.4,Eq.41(http://earthinfo.nga.mil/GandG/publications/tr8350.2/wgs84fin.pdf)

12. Eq.2.57inMITEssentialsofGeophysicsOpenCourseWarenotes(http://ocw.mit.edu/courses/earthatmosphericandplanetarysciences/12201essentialsofgeophysicsfall2004/lecturenotes/ch2.pdf)

13. JohnP.Vinti,GimJ.Der,NinoL.Bonavito(1998).OrbitalandCelestialMechanics.Progressinastronauticsandaeronautics,v.177.AmericanInstituteofAeronauticsandAstronautics.p.171.ISBN1563472562.

14. ArthurGordonWebster(1904).TheDynamicsofParticlesandofRigid,Elastic,andFluidBodies:beinglecturesonmathematicalphysics.B.G.Teubner.p.468.

15. IsaacNewton:PrincipiaBookIIIPropositionXIXProblemIII,p.407inAndrewMottetranslation.16. SeethePrincipiaonlineatAndrewMotteTranslation

(http://ia310114.us.archive.org/2/items/newtonspmathema00newtrich/newtonspmathema00newtrich.pdf)17. Table1.1IERSNumericalStandards(2003)(ftp://tai.bipm.org/iers/convupdt/chapter1/icc1.pdf))18. AverilB.Chatfield(1997).FundamentalsofHighAccuracyInertialNavigation.Volume174inProgressin

AstronauticsandAeronautics.AmericanInstituteofAeronauticsandAstronautics.Chapter1,PartVIIIp.7.ISBN1563472430.

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