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{{for|the interpretation of this theorem in terms of symmetry of second derivatives of a mapping <math>f \colon \mathbb{R}^n \to \mathbb{R}</math> |Symmetry of second derivatives}}
[[File:Elipsoid zplostely.png|thumb |200px |Figure 1: An ellipsoid]]
[[File:Gnuplot ellipsoid.svg|thumb|200px|Figure 2: Wireframe rendering of an ellipsoid (oblate spheroid)]]
'''Clairaut's theorem''', published in 1743 by [[Alexis Clairaut|Alexis Claude Clairaut]] in his ''Théorie de la figure de la terre, tirée des principes de l'hydrostatique'',<ref name=RoyalSoc>[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=an0JW-H3C8&sig=BMkuXfZEsK3p0tzrZ1Jvfcy7hmw&hl=en&sa=X&oi=book_result&resnum=10&ct=result From the catalogue of the scientific books in the library of the Royal Society.]</ref> synthesized physical and geodetic evidence that the Earth is an oblate rotational [[ellipsoid]].<ref name= Torge>{{cite book |title=Geodesy: An Introduction |edition=3rd |author=Wolfgang Torge |page=10 |url=http://books.google.com/books?id=pFO6VB_czRYC&pg=PA109&dq=%22Clairaut%27s+theorem%22&lr=&as_brr=0&sig=ACfU3U34GaPhl4tA9duUMLQpm77hiKb-RQ#PPA10,M1 |isbn=3-11-017072-8 |year=2001 |publisher=Walter de Gruyter }}</ref><ref name=Routh>{{cite book |author=Edward John Routh |title=A Treatise on Analytical Statics with Numerous Examples |page=154 |year=2001|isbn=1-4021-7320-2 |publisher=Adamant Media Corporation |volume=Vol. 2 |url=http://books.google.com/books?id=yKmdk4LZxhMC&pg=RA1-PA40&dq=isbn=1-4021-7320-2&sig=ACfU3U2uhAKDJtIYZEY-Jf-1e5wf7UgG1w#PPA154,M1 }} A reprint of the original work published in 1908 by Cambridge University Press.</ref> It is a general mathematical law applying to spheroids of revolution. It was initially used to relate the gravity at any point on the Earth's surface to the position of that point, allowing the [[ellipticity]] of the Earth to be calculated from measurements of gravity at different latitudes.

==Formula==
Clairaut's formula for the acceleration due to gravity ''g'' on the surface of a spheroid at latitude φ, was:<ref name=Ball>[http://www.maths.tcd.ie/pub/HistMath/People/Clairaut/RouseBall/RB_Clairaut.html W. W. Rouse Ball ''A Short Account of the History of Mathematics'' (4th edition, 1908)]</ref><ref name=Rouse2>{{cite book |title=A short account of the history of mathematics |author= Walter William Rouse Ball |page=384 |url=http://books.google.com/books?id=O-UGAAAAYAAJ&dq=A+Short+Account+of+the+History+of+Mathematics'+(4th+edition,+1908)+by+W.+W.+Rouse+Ball.&pg=PP1&ots=327JhZ192M&sig=w-HWPhOnc6JAlzlMoralry7rIL4&hl=en&sa=X&oi=book_result&resnum=1&ct=result#PPA384,M1
|year=1901 |publisher=Macmillan |edition=3rd }}</ref>

:<math> g = G \left[ 1 + \left(\frac{5}{2} m - f\right) \sin^2 \phi \right] \ , </math>

where ''G'' is the value of the acceleration of gravity at the equator, ''m'' the ratio of the centrifugal force to gravity at the equator, and ''f'' the [[flattening]] of a [[meridian (geography)|meridian]] section of the earth, defined as:
:<math>f = \frac {a-b}{a} \ , </math>
(where ''a'' = semimajor axis, ''b''=semiminor axis ).

Clairaut derived the formula under the assumption that the body was composed of concentric coaxial spheroidal layers of constant density.<ref>{{cite book
| last = Poynting
| first = John Henry
| authorlink =
| coauthors = Joseph John Thompson
| title = A Textbook of Physics, 4th Ed.
| publisher = Charles Griffin & Co.
| year = 1907
| location = London
| pages = 22–23
| url = http://books.google.com/books?id=TL4KAAAAIAAJ&pg=PA22
| doi =
| id =
| isbn = }}</ref>
This work was subsequently pursued by [[Pierre-Simon Laplace|Laplace]], who relaxed the initial assumption that surfaces of equal density were spheroids.<ref name=Todhunter>{{cite book |author=Isaac Todhunter |title=A History of the Mathematical Theories of Attraction and the Figure of the Earth from the Time of Newton to that of Laplace |volume=Vol. 2 |publisher=Elibron Classics |isbn=1-4021-1717-5 |url=http://books.google.com/books?id=blZ_Tar9IRMC&pg=RA1-PA500&dq=%22Clairaut%27s+theorem%22&lr=&as_brr=0&sig=ACfU3U0BK0IMg4DPZTFon_yf_DyT4wOlcQ#PPA62,M1 }} Reprint of the original edition of 1873 published by Macmillan and Co.</ref>
[[Sir George Stokes, 1st Baronet|Stokes]] showed in 1849 that the theorem applied to any law of density so long as the external surface is a spheroid of equilibrium.<ref name=Fisher>{{cite book |title=Physics of the Earth's Crust |author=Osmond Fisher |page=27 |url=http://books.google.com/books?id=o8oPAAAAIAAJ&pg=PA27&dq=%22Clairaut%27s+theorem%22&lr=&as_brr=0
|year=1889 |publisher=Macmillan and Co. }}</ref><ref name= Poynting>{{cite book |title=A Textbook of Physics |author= John Henry Poynting & Joseph John Thomson |url=http://books.google.com/books?id=TL4KAAAAIAAJ&pg=PA23&dq=%22Clairaut%27s+theorem%22&lr=&as_brr=0#PPA22,M1 |page=22 |year=1907 |publisher=C. Griffin }}</ref> A history of the subject, and more detailed equations for ''g'' can be found in Khan.<ref name=Khan>[http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19690003446_1969003446.pdf NASA case file ''On the equilibrium figure of the earth'' by Mohammad A. Khan (1968)]</ref>

The above expression for ''g'' has been supplanted by the Somigliana equation:
:<math>g = G \left[ \frac{1+k\sin^2 \phi}{\sqrt{1-e^2 \sin^2 \phi }} \right] \ , </math>
where, for the Earth, G =9.7803267714 ms<sup>−2</sup>; k =0.00193185138639 ; e<sup>2</sup> =0.00669437999013.<ref name=Somigliana>[http://ocw.mit.edu/NR/rdonlyres/Earth--Atmospheric--and-Planetary-Sciences/12-201Fall-2004/E7A9DF78-ADC6-49A7-8812-1D8244939398/0/ch2.pdf Eq. 2.57 in MIT Earth Atmospheric and Planetary Sciences OpenCourseWare notes]</ref>

==Clairaut's relation==
{{Main|Clairaut's relation}}

A formal mathematical statement of the (unrelated) Clairaut's theorem is:<ref name=Pressley>{{cite book |author=Andrew Pressley |title=Elementary Differential Geometry |page=183 |url=http://books.google.com/books?id=UXPyquQaO6EC&pg=PA185&dq=%22Clairaut%27s+theorem%22&lr=&as_brr=0&sig=ACfU3U214J0zkWQcRXLTohVjdHUD3Fuk2A#PPA183,M1
|isbn=1-85233-152-6 |publisher=Springer |year=2001 }}</ref>
{{quotation|Let γ be a [[geodesic]] on a [[surface of revolution]] ''S'', let ρ be the distance of a point of ''S'' from the [[axis of rotation]], and let ψ be the angle between γ and the [[Meridian (geography)|meridians]] of ''S''. Then ρ sin ψ is constant along γ. Conversely, if ρ sin ψ is constant along some curve γ in the surface, and if no part of γ is part of some parallel of ''S'', then γ is a geodesic.|Andrew Pressley: ''Elementary Differential Geometry'', p. 183}}

Pressley (p. 185) explains this theorem as an expression of conservation of angular momentum about the axis of revolution when a particle slides along a geodesic under no forces other than those that keep it on the surface.

==Geodesy==
The spheroidal shape of the Earth is the result of the interplay between [[gravity]] and [[centrifugal force]] caused by the Earth's rotation about its axis.<ref name=Vinti>{{cite book |title=Orbital and Celestial Mechanics |series=Progress in astronautics and aeronautics, v. 177 |author=John P. Vinti, Gim J. Der, Nino L. Bonavito |page=171 |url=http://books.google.com/books?id=-dXzdYHvPgMC&pg=PA172&dq=Earth+spheroid+centrifugal+date:1990-2008&lr=&as_brr=0&sig=ACfU3U0YCa9N8606CejuHyuolKmh56JOtw#PPA171,M1 |isbn=1-56347-256-2 |year=1998 |publisher=American Institute of Aeronautics and Astronautics}}</ref><ref name=Webster>{{cite book |title=The Dynamics of Particles and of Rigid, Elastic, and Fluid Bodies: being lectures on mathematical physics |author=Arthur Gordon Webster |year=1904 |publisher=B.G. Teubner |url=http://books.google.com/books?id=2kMNAAAAYAAJ&printsec=titlepage#PPA468,M1 |page=468 }}</ref> In his ''Principia'', [[Isaac Newton|Newton]] proposed the equilibrium shape of a homogeneous rotating Earth was a rotational ellipsoid with a flattening ''f'' given by 1/230.<ref name=Newton>Isaac Newton: ''Principia'' Book III Proposition XIX Problem III, p. 407 in Andrew Motte translation.</ref><ref name=Principia>See the ''Principia'' on line at [http://ia310114.us.archive.org/2/items/newtonspmathema00newtrich/newtonspmathema00newtrich.pdf Andrew Motte Translation]</ref> As a result gravity increases from the equator to the poles. By applying Clairaut's theorem, [[Pierre-Simon Laplace|Laplace]] was able to deduce from 15 gravity values that ''f'' = 1/330. A modern estimate is 1/298.25642.<ref>[ftp://tai.bipm.org/iers/convupdt/chapter1/icc1.pdf Table 1.1 IERS Numerical Standards (2003)])</ref> See [[Figure of the Earth]] for more detail.

For a detailed account of the construction of the [[Reference ellipsoid|reference Earth model]] of geodesy, see Chatfield.<ref name=Chatfield>{{cite book |title= Fundamentals of High Accuracy Inertial Navigation |url=http://books.google.com/books?id=2hJTDpT2U1UC&pg=PA1&dq=frame+coordinate+%22state+of+motion%22&lr=&as_brr=0&sig=ACfU3U2NOYvih-VaDyv1CxAkTc7L1AaRXQ#PPA7,M1
|isbn=1-56347-243-0 |year=1997 |author=Averil B. Chatfield |publisher=American Institute of Aeronautics and Astronautics |series=Volume 174 in ''Progress in Astronautics and Aeronautics'' |nopp= true |pages= Chapter 1, Part VIII p. 7 }}</ref>

==References==
<references/>

[[Category:Geodesy]]
[[Category:Global Positioning System]]
[[Category:Navigation]]
[[Category:Surveying]]
[[Category:Physics theorems]]
[[Category:Gravimetry]]

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