J 28 LAMBKET: COKSTITUTIOK OF TllS EARTH 



value, and the upper and lower limits are so close together and so 

 close to 1/297.2 that our determinations of the flattening from pen- 

 dulum obser\'ations or from triangulaticn are not accurate enough 

 for us to say that one law of density represents observed facts 

 better than another.^ 



We can take hold of the matter by the other end. Let us 

 assume as the data of observation the values of the flattening, of 

 {C~A)/C, of the mean density and sm'f ace density of the earth, 

 and of the ratio of the centrifugal force of rotation to gravity at 

 the equator, a quantity whose value has already been tacitly 

 assumed in our previous discussions. Let us see what conclu- 

 sions about the density are allowable. We are still supposing 

 hydrostatic equilibrium, and for this to be stable, densities must 

 increase with depth; let us further suppose that the density 

 changes continuously and that the rate of increase diminishes 

 as the depth increases. The limits of density shown in table 2 

 have been derived by Stieltjes.^ 



The data assumed as the basis of table 2 are not quite the same 

 as for the previous table, but the difference is of little conse- 

 quence. 



Before leaving the subject of densities, Wiechert's hypothesis 

 should be mentioned. Legendre's law of densities and others 



* The flattenings so far given have all been computed from formulas that are 

 accurate only to small quantities of the first order in the ellipticity and the ratio 

 of the centrifugal force at the equator to gravity there. Since these quantities are, 

 respectively, about 1/297 ^nd 1/289, it is seen that the terms of the second order, 

 7. e., in the squares and products of these quantities, might very well affect the tenths 

 or even tlie units in the reciprocal of the flattening. It happens, however, that in 

 determining the flattening from the law of density and the rate of rotation, the effect 

 of the terms of the second order is small, i. e., a few tenths only, and the general con- 

 clusions hold good as stated. See reference to Veronnet in previous footnote; 

 also Darwin, The theory of the figure of the earth carried to the second order of small 

 quantities. Monthly Notices of the Royal Astronomical Society 60: 82. 1900. 

 Scientific Papers 3: 79. In determining the flattening from pendulum observations 

 the terms of the second order have a somewhat greater effect. 



' Archives neerlandaises (Haarlem, 1884) 19: 456. See also Tisser-'^nd, MS- 

 caniqne Celeste, 2: 227. 



