ON THE MAGNITUDE OF THE SOLAR SYSTEM. 103 



but it is usual to assume that the deeper matter is distributed, accord- 

 ing to Lagrange'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 nutation, we 

 readily obtain the flattening. As yet these methods do not give con- 

 sistent 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 comi^ute the function of the 

 earth's moments of inertia which we have just been considering, 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 comi)aring 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 pendu- 

 lums at the surface and down in mines; and in the (;ase of known 

 masses of metal they have been made with torsion balances, fine chem- 

 ical balances, and pendulums. The surface density results from a study 

 of the materials composing the earth's crust, but notwithstanding the 

 api)arent 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 i)oint out 

 that the luni-solar i^recession can not be directly observed, but must be 

 derived from the general precession. The former of these qualities 

 depends only upon the action of the sun and moon, while the latter is 

 affected in addition by the action of all the planets, and to ascertain 

 what that is we must determine their masses. The methods of doing 

 so fall into two great classes, according as the planets dealt with have 

 or have not satellites. The most favorable case is that in which one or 

 more satellites are present, because the mass of the primary follows 

 immediately from their distances and revolution times; but even then 

 there is a difiticulty in the way of obtaining very exact results. By 

 extending the observations over sufficiently long periods the revolution 

 times may be ascertained with any desired degree of accuracy; but all 

 measurements of the distance of a satellite from its primarj- are 

 affected by personal equation, which we can not be sure of completely 

 eliminating, and thus a considerable margin of uncertainty is brought 

 into the masses. In the cases of Mercury and Venus, which have no 

 satellites, and to a certain extent in the case of the earth also, the only 

 available way of ascertaining the masses is from the perturbations 

 produced by the action of the various planets on each other. These 

 l^erturbations are of two kinds, j)eriodic and secular. When sufficient 

 data have been accumulated for the exact determination of the secular 

 pertnrbatioriS they will give the best results, but as yet it remains 

 advantageous to employ the periodic perturbations also. 



