PREMONITIONS OF ISOSTASY 175 



Though no explicit assumption was made by Stokes as to the distri- 

 bution of density, there is none the less a very important implicit postu- 

 late. It requires no analysis to show that if there were very great 

 irregularities in the distribution of density the level surfaces or equipo- 

 tentials of the globe would not necessarily be approximately spherical, 

 and it is essential to make some estimate of the degree of irregularity 

 implied in Stokes's theorem. 



Clairaut's theorem is expressed in terms of the flattening or ellipticity 

 of the earth, known to be about 1/298 (though possibly it is a little 

 larger), and is denoted by e.^- If the mean radius of the earth is taken 

 as unity, e is 1/298 of a radian, or 11' 32" = 692". Stokes neglects all 

 terms in e^, and e^ is equivalent to 2". 3. Now if the ellipsoid of revolu- 

 tion whose ellipticity is e is an equipotential surface, it is easy to prove 

 that the maximum angle between the normal to the equipotential surface 

 and the radius vector is e. Stokes's theorem is consequently true only of 

 an earth on which the angle between the geocentric vector and the normal 

 to the geoid does not exceed a quantity of the same order as e. Otherwise 

 expressed, its truth is limited to cases in which the deflection of the ver- 

 tical is of the order of e-. Now observation shows that a deflection of 

 23" is, relatively speaking, very large. Mr. Hayford in his first memoir, 

 which will presently be noticed more at length, records the observed de- 

 flection at 509 stations, and only at 7 of them does this exceed 20", the 

 highest approaching 30", which is also about the maximum observed in 

 any country. At seven-eighths of these stations the deflection is less 

 than 10". 



Thus, closely enough, the assumption implied in Stokes's theorem may 

 be said to be that 



e + 23" = e + e/30 = 1.033 e 



is to be regarded as of the same order of magnitude as e or, since 



23" = 10 e-, 



substituting the value of r and squaring each side of this equation reduces it to the re- 

 quired algebraic form. E is the earth's mass, e the ellipticity of the meridian, a the 

 mean radius of the earth's surface, ;• the geocentric radius vector, and m the ratio of 

 centrifugal force at. the equator to gravity at the equator. 



The geoid Is a far more complex solid, being one in which all irregularities due to 

 local attraction are superposed upon the surface of the tenth degree. The maximum de- 

 parture of the geoid from the theoretical spheroid is supposed to be about 100 meters 

 and to occur under "the roof of tlie world" in central Asia, 



"Stokes extends the application of the term ellipticity to a slightly irregular figure, 

 such as the geoid, or sealevel surface. At a distance from the eartli such as that of the 

 moon, the attraction of the geoid would coincide with that of the mean ellipsoid; conse- 

 quently the mean flattening of the geoid may be regarded as the ellipticity of the mean 

 ellipsoidal spheroid. 



