W. Harkness — Magnitude of the Solar System. 241 



the resj)ective points as seen from the moon, they afford a 

 direct measure of the flattening. The second and third astro- 

 nomical methods are based upon certain perturbations of the 

 moon which depend upon the figure of the earth, and should 

 give extremely accurate results, but unfortunately very great 

 difficulties oppose themselves to the exact measurement of 

 the perturbations. There is also an astronomico-geological 

 method which can not yet be regarded as conclusive on 

 account of our lack of knowledge respecting the law of density 

 which prevails in the interior of the earth. It is based upon 

 the fact that a certain function of the earth's moments of iner- 

 tia can be determined from the observed values of the coeffi- 

 cients of precession and nutation, and could also be determined 

 from the figure and dimensions of the earth if we knew the 

 exact distribution of matter in its interior. Our present 

 knowledge on that subject is limited to a superficial layer not 

 more than ten miles thick, but it is usual to assume that the 

 deeper matter is distributed according to La Grange's law, and 

 then by writing the function in question in a form which 

 leaves the flattening indeterminate, and equating the expres- 

 sion so found to the value given by the precession and nuta- 

 tion, we readily obtain the flattening. As yet these six 

 methods do not give consistent results, and so long as serious 

 discrepancies remain between them there can be no security 

 that we have arrived at the truth. 



It should be remarked that in order to compute the function 

 of the earth's moments of inertia which we have just been con- 

 sidering, we require not only the figure and dimensions of the 

 earth and the law of distribution of density in its interior, but 

 also its mean and surface densities. The experiments for 

 determining the mean density have consisted in comparing the 

 earth's attraction with the attraction either of a mountain, or 

 of a known thickness of the earth's crust, or of a known mass 

 of metal. In the case of mountains the comparisons have been 

 made with plumb lines and pendulums ; in the case of known 

 layers of the earth's crust they have been made by swinging- 

 pendulums at the surface and down in mines ; and in the case 

 of known masses of metal they have been made with torsion 

 balances, fine chemical balances and pendulums. The surface 

 density results from a study of the materials composing the 

 earth's crust, but notwithstanding the apparant simplicity of 

 that process, it is doubtful if we have yet attained as accurate 

 a result as in the case of the mean density. 



Before quitting this part of our subject, it is important to 

 point out that the luni-solar precession can not be directly 

 observed, but must be derived from the general precession. 

 The former of these quantities depends only upon the action 



