Editing Clairaut's theorem (gravity) (section)

==Formula==
Clairaut's theorem says that the acceleration due to gravity ''g'' (including the effect of centrifugal force) on the surface of a spheroid in [[hydrostatic equilibrium]] (being a fluid or having been a fluid in the past, or having a surface near sea level) at latitude {{mvar|φ}} is:<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=[https://archive.org/details/ashortaccounthi00ballgoog/page/n359 384] | url=https://archive.org/details/ashortaccounthi00ballgoog | quote=A Short Account of the History of Mathematics' (4th edition, 1908) by W. W. Rouse Ball. |year=1901 |publisher=Macmillan |edition=3rd }}</ref>

<math display="block"> g(\varphi) = G_e \left[ 1 + \left(\frac{5}{2} m - f\right) \sin^2 \varphi \right] \, , </math>where 

* <math>G_e</math> is the value of the acceleration of gravity at the equator, 
* <math>m</math> the ratio of the centrifugal force to gravity <math display="inline">\omega^2R_E(\phi=0)/g(\phi=0)</math> at the equator, and 
* <math>f</math> the [[flattening]] of a [[meridian (geography)|meridian]] section of the earth, defined as:

<math display="block">f = \frac {a-b}{a} \, , </math>(where ''a'' = semi-major axis, ''b'' = semi-minor axis).

The contribution of the centrifugal force is approximately <math display="inline">-G_e m\cos^2\varphi,</math> whereas gravitational attraction itself varies approximately as <math display="inline">G_e\left(\frac32 m-f\right)\sin^2\varphi.</math> This formula holds when the surface is perpendicular to the direction of gravity (including centrifugal force), even if (as usually) the density is not constant. (In which case the gravitational attraction can be calculated at any point from the shape alone, without reference to <math>m</math>.) For the earth, <math display="inline">m\approx 1/289,</math> and <math display="inline">\frac52 m\approx 1/116,</math> while <math display="inline">f\approx 1/300,</math> so <math>g</math> is greater at the poles (5.3‰, 0.43% Newton) than on the equator.<ref name="Stokes" />

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 |author2=Joseph John Thompson | title = A Textbook of Physics | edition = 4th | publisher = Charles Griffin & Co. | year = 1907 | location = London | pages = [https://archive.org/details/atextbookphysic01thomgoog/page/n32 22]–23 | url = https://archive.org/details/atextbookphysic01thomgoog }}</ref> This work was subsequently pursued by [[Pierre-Simon Laplace|Laplace]], who assumed surfaces of equal density which were nearly spherical.<ref name="Stokes" /><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 |date=January 1999 |volume=2 |publisher=Elibron Classics |isbn=1-4021-1717-5 |url=https://books.google.com/books?id=blZ_Tar9IRMC&dq=%22Clairaut%27s+theorem%22&pg=PA62 }} Reprint of the original edition of 1873 published by Macmillan and Co.</ref> The English mathematician [[Sir George Stokes, 1st Baronet|George Stokes]] showed in 1849<ref name="Stokes">{{cite journal |last1=Stokes |first1=G. G. |title=On attractions, and on Clairaut's theorem |journal=The Cambridge and Dublin Mathematical Journal |date=1849 |volume=4 |pages=194–219 |url=https://archive.org/stream/cambridgeanddub03unkngoog#page/n201/mode/2up}}</ref> 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=https://books.google.com/books?id=o8oPAAAAIAAJ&dq=%22Clairaut%27s+theorem%22&pg=PA27 |year=1889 |publisher=[[Macmillan and Co.]] }}</ref><ref name="Poynting">{{cite book |title=A Textbook of Physics |author1=John Henry Poynting |author2=Joseph John Thomson |url=https://archive.org/details/bub_gb_TL4KAAAAIAAJ |quote=Clairaut's theorem. |page=[https://archive.org/details/bub_gb_TL4KAAAAIAAJ/page/n32 22] |year=1907 |publisher=C. Griffin }}</ref> A history of more recent developments and more detailed equations for ''g'' can be found in Khan.<ref name="Khan">Mohammad A. Khan (1968) [https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19690003446_1969003446.pdf NASA case file ''On the equilibrium figure of the earth'']. ''ntrs.nasa.gov/archive'' [https://web.archive.org/web/20100528083748/https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19690003446_1969003446.pdf Archived] on 2010-05-28.</ref>

The above expression for ''g'' has been supplanted by the [[Somigliana equation]] (after [[Carlo Somigliana]]).