THE ROTATION-PERIOD OF A HETEROGENEOUS 



SPHEROID. 



It is a simple problem to determine the rotation-period of an ellipsoid 

 of revolution, if it be postulated that the density of the body is uniform, 

 and that the form is that assumed by a perfect liquid under like conditions 

 of rotation. A table of the rotation-periods of such a body having the same 

 volume and mean density as that of the earth, computed for various values 

 of the eccentricity of an elHptic meridian section, will be found on p. 327 of 

 Part II of Thomson and Tait's Natural Philosophy (edition of 1890). It 

 is the purpose of the present investigation to obtain analogous results for 

 an ellipsoid of variable density, assuming a law of increasing density from 

 surface to center approximate to that actually possessed by the present 

 earth. The law of density assumed in the computation is the well-known 

 law of Laplace: 



, = QilB^ (1) 



in which the symbols have the meaning given on page 64. According 

 to this law the internal layers or shells of equal density gradually change 

 from the shape of the surface to forms more and more nearly spherical 

 as the center of the spheroid is approached. The forms of these layers 

 are best expressed in the case of ellipsoids of revolution by the ellipticity 

 of a meridional section. This number is computed by subtracting the 

 length of the polar or short axis from the length of the equatorial or long 

 axis and dividing the result by the length of the equatorial axis.^ The 

 variation in the value of the ellipticities is shown by the dotted line in 

 fig. 7. In this diagram the polar axis is represented as divided into ten 

 equal parts. The elhpticities of the shells of equal density are expressed 

 as percentages of the ellipticity of the surface. Thus the ellipticity of the 

 shell that cuts the polar axis at 0.5 of the distance from the center to the 

 surface is equal to 85 per cent of the surface ellipticity, while the ellipticity 

 of the central shell is about 80 per cent of the surface ellipticity. 



If we assume a surface density of 2.75 and a mean density of 5.50, the 

 above expression takes the form: 



4.365 ao . 2.4605 a 



— " sm (2) 



a Go 



The variation in density according to this law is shown graphically by 

 the continuous curve of fig. 7. 



^ The mean axis or the polar axis is often used as the divisor. There is little difference 

 in the results for the small values of the ellipticity usually involved. 



63 



