THE ROTATION-PERIOD OF A HETEROGENEOUS SPHEROID. 65 



b-c 

 £ = -r— = ellipticity of any homogeneous internal shell of spheroid. 



eo = ellipticity of surface of spheroid. 

 M = centripetal acceleration at equator of spheroid. 



While the Laplacian law of density originated in an assumption of 

 Laplace which had little to recommend it beyond mere plausibility, the 

 law is now believed to be fairly close to the truth. The computed values 

 of the earth's precession based upon this law of density agree well with 

 the observed values. The law is probably quite as near to the truth as is 

 the measured value of the earth's mean density, which must enter as a 

 basal number into any formula of density we may adopt. 



The plan of the investigation is substantially as follows: The attrac- 

 tion of the heterogeneous spheroid of given ellipticity is first found for 

 points on the equator and at the poles of the spheroid. These results are 

 substituted in Clairaut's well-known equation connecting gravity at the 

 pole and at the equator with the equatorial centripetal acceleration, and 

 hence with the rotation-period of the earth. In this manner the rotation- 

 period for any given ellipticity of meridional section becomes known. 



Clairaut himself gave expression to formulas which give the attraction 

 at external points of any rotating liquid ellipsoid.* For polar and equa- 

 torial points these may be written as follows (referring to the preceding list 

 for the meaning of the symbols) : 



Equatorial i i % rdo s 



attraction = a, = A ~, K -] / pda^ + A..^! pd(ah) I (3) 



Polar 

 attraction = ap=i-.i^K-j / pda'--^^-^l pd{ah) [ (4) 



•^/;.(a.)} 



In both formulas K is a constant whose value depends upon the units 

 of measure in which the various magnitudes are expressed. 



These expressions assume that each stratum of density p has the 

 ellipticity e that would exist for the given rotation-period if all strata 

 were perfectly liquid. In other words, the formula is built upon the hypoth- 

 esis of the perfect fluidity of the spheroid. If we apply these expressions 

 to the present earth we assume that the rigidity of the earth is not sufficient 

 to withstand for geological intervals of time the stresses that would exist 

 if the form of its surface differed materially from that of a free liquid. 



The above expressions (3) and (4) can not be integrated until we sub- 

 stitute for p the appropriate law of density from (1) above. Using the 

 notation : 



<p{o)='^ (sin qa^-qa^ cos qa^) (5) 



* See "History of the theories of attraction and the figure of the earth," by I. Todhunt«r 

 London, 1873, vol. 1, p. 220. 



