D' It is thu3 seen that k is not a mere number, but a quantity such that the second member of the equation is force as well as the first member. k is, in fact, the cube of a length divided by the product of a mass and the square of a time.
Denoting the mean density of the earth by p, it is found in the paper referred to that kp36797 X / ( second , this quantity being the reciprocal of the square of a time. In a more interesting, as well as a more intelligible, form this relation may be written where it is the number 3.1415+ and T is the time it would take an infinitesimal satellite to pass around the earth under the law of gravi tation, just grazing the equator, if there were no atmosphere to impede its progress. This time is seen to be 1 hour, 24 minutes, 20.9 seconds.
It is clear then that p will be given by the above equation if k is known. This quantity has been measured directly by several observers. A mean of the most recent and most trustworthy determination, is in C.G.S. units (see (The Century's Progress in Applied Mathematics,' Science, N.S., Vol. XI), k=6673 X 10' Thus p is found to be 5.514 times the density of pure water. The uncertainty of this value as shown by the computation is about four units in the last place of decimals. In view of this fact, and of the great difficulties in measuring the quantity k, we shall use the round number 5.5 as hitherto. • From the volume of the earth given above the following results are derived for its mass : = 1.311 X pounds Mass of earth --= 6,552 X tons (of 2,000 pounds) 5,945 X kilograms For convenience of reference the masses that have been given above are here collected and expressed in tons of 2,000 pounds.
Mass of earth = 6,550 X Mass of centrosphere = 6,529 X 10* Mass of lithosphere = 21 X Mass of hydrosphere = 1.49 X 101* Mass of atmosphere — 7 X 10" The Internal Heat of the Earth.— That the earth has a great store of heat a few miles be low the surface is amply proved by a variety of geological phenomena. Wherever deep wells, or bore holes, have been sunk the temperature is observed to increase with depth at a rate of about a degree F. for 60 feet. This shows that heat is being conducted from the interior to the surface of the earth and is thence radiated into surrounding space. The amount of this heat is sufficient to melt a layer of ice six to eight milli meters (or one-fourth inch, say) thick, covering the globe, per annum, or more than 800 cubic miles of ice. It seems most probable (though it is by no means certain) that the temperature of the centrosphere is high enough to melt all known rocks, although the lower strata of the crust do not assume the molten or viscid form of lavas except on relief from the great pres sure to which they are subject. This was, es sentially, investigate the view of Fourier, the earliest physi cist to nvestigate this question, and it is still regarded as the most plausible hypothesis with respect to the actual state of the earth.
Whether this hypothesis is exact or not, how ever, the theory of heat conduction founded by Fourier enables us to draw with certainty two important conclusions with respect to the earth, namely: (a) that the heat conducted from the interior to the surface escapes as if there were neither atmosphere nor oceans; and (b) that a million years is the smallest unit of time con venient for measuring the historical succession of thermal events.
With less certainty it may be affirmed that the earth is shrinking in bulk as it cools, and that this is one of the principal causes of the grand crust crumplings and volcanic activities to which geologists pay special attention. This cubical contraction goes on exceedingly slowly, however; requiring, so far as one can infer from present indications, nothing short of thousands of millions of years for its completion.
The Earth as a Time The time of rotation of the earth is the most trustworthy unit of time man has discovered. The present value of this unit is 86164.1 mean solar seconds. That the earth must rotate with great steadi ness is at once apparent when the immense amount of its energy of rotation is considered. This energy is about 156X10" foot-pounds, or 2X10" ergs. It is more than will be developed at Niagara (at the present rate of 5,000,000 horse power, say) in a million million years. Nevertheless, the period of rotation of the earth is subject to variation from four obvious causes. These are (1), secular contraction of the earth's mass; (2), the influx of meteorites, or meteoric dust; (3), tidal friction; and (4), shiftings in position of the surface load of the earth, as in the processes of sedimentation, glaciation, etc.
The effects of the first two causes have been investigated. (Astronomical Journal, No. 502). Contraction tends to shorten the day, and may possibly cause a diminution of as much as 6 per cent of the present length. The change goes on very slowly, however, and will not be per ceptible in any such interval as that of human history (20 centuries, say). Meteoric dust tends to lengthen the day, but at the present rate of influx (about 20,000,000 meteorites daily) the effect will not amount to so much as a quarter of a second in less than a million million years. During this interval of time the total effect, substantially, from• secular contraction will have accrued.
The effects of the other causes named have not been evaluated, though the last one is prob ably the most important of all of them ; since it may be easily shown that such shiftings of the surface load as are now taking place on the earth may modify the length of the day by an amount which, if cumulative, might in a few centuries seriously disturb astronomical reckon ings.
'Mecanique Celeste' and 'Systeme du Monde' ; Todhunter, 'His tory of the Theories of Attraction and the Figure of the Earth' (1873) ; Clarke, 'Geodesy' (I&O) ; Helmert, 'Die Mathematischen and Physikalischen Theorieen der Hiiheren Geodasie' (1880, Vol. I; 1884, Vol. II) ; Wood ward, R. S., 'Smithsonian Geographical Tables' (published by the Smithsonian Institution, Washington 1894) ; Thomson (Lord Kelvin) and Tait, 'Treatise on Natural Philosophy' (1890).