Moduli of resistance to compression and of rigidity. Hence, for an isotropic solid, (18) gives the expressions which we have used above, (12) of § 673. (m.) To interpret the coefficients m and n in connexion with elementary ideas as to the elasticity of the solid; first let a = b = c = 0, and e=f=g=18: in other words, let the substance experience a uniform dilatation, in all directions, producing an expansion of volume from 1 to 1 + 8. In this case (18) becomes Hence (mn) 8 is the normal force per unit area of its surface required to keep any portion of the solid expanded to the amount specified by d. Thus min measures the elastic force called out by, or the elastic resistance against change of volume: and viewed as a modulus of elasticity, it may be called the bulkmodulus. [Compare $$ 692, 693, 694, 688, 682, and 680.] What is commonly called the "compressibility" is measured by 1/(m-n). And let next e=f=g=b=c= 0; which gives w = {na2; and, by (19), S=na. This shows that the tangential force per unit area required to produce an infinitely small shear (§ 171), amounting to a, is na. Hence n measures the innate power of the body to resist change of shape, and return to its original shape when force has been applied to change it: that is to say, it measures the rigidity of the substance. Appendix D. Dissipation of energy disregarded by many followers of Hutton. (D).—ON THE SECULAR COOLING OF THE EARTH*. (a.) For eighteen years it has pressed on my mind, that essential principles of Thermo-dynamics have been overlooked by those geologists who uncompromisingly oppose all paroxysmal hypotheses, and maintain not only that we have examples now before us, on the earth, of all the different actions by which its crust has been modified in geological history, but that these actions have never, or have not on the whole, been more violent in past time than they are at present. * Transactions of the Royal Society of Edinburgh, 1862 (W. Thomson). of energy solar (b.) It is quite certain the solar system cannot have gone on, Dissipation even as at present, for a few hundred thousand or a few million from the years, without the irrevocable loss (by dissipation, not by annihilation) of a very considerable proportion of the entire energy initially in store for sun heat, and for Plutonic action. It is quite certain that the whole store of energy in the solar system has been greater in all past time than at present; but it is conceivable that the rate at which it has been drawn upon and dissipated, whether by solar radiation, or by volcanic action in the earth or other dark bodies of the system, may have been nearly equable, or may even have been less rapid, in certain periods of the past. But it is far more probable that the secular rate of dissipation has been in some direct proportion to the total amount of energy in store, at any time after the commencement of the present order of things, and has been therefore very slowly diminishing from age to age. climate in the pro sun of a few million years ago. (c.) I have endeavoured to prove this for the sun's heat, in an Terrestrial article recently published in Macmillan's Magazine (March 1862)*, fluenced by where I have shown that most probably the sun was sensibly bably hotter hotter a million years ago than he is now. Hence, geological speculations assuming somewhat greater extremes of heat, more violent storms and floods, more luxuriant vegetation, and hardier and coarser grained plants and animals, in remote antiquity, are more probable than those of the extreme quietist, or "uniformitarian" school. A middle path, not generally safest in scientific speculation, seems to be so in this case. It is probable that hypotheses of grand catastrophes destroying all life from the earth, and ruining its whole surface at once, are greatly in error; it is impossible that hypotheses assuming an equability of sun and storms for 1,000,000 years, can be wholly true. (d.) Fourier's mathematical theory of the conduction of heat is a beautiful working out of a particular case belonging to the general doctrine of the "Dissipation of Energyt." A characteristic of the practical solutions it presents is, that in each case a * Reprinted as Appendix E, below. + Proceedings of Royal Soc. Edin., Feb. 1852. "On a universal Tendency in Nature to the Dissipation of Mechanical Energy," Mathematical and Physical Papers, by Sir W. Thomson, 1882, Art. LIX. Also, "On the Restoration of Energy in an unequally Heated Space," Phil. Mag., 1853, first half year, Mathematical and Physical Papers, by Sir W. Thomson, 1882, Art. LXII. Mathema- Criterion of an es- of heat in a now applied to estimate date of earth's consolidation, from data of present underground temperature. Value of local geothermic surveys, for estimation of absolute dates in geology. distribution of temperature, becoming gradually equalized through an unlimited future, is expressed as a function of the time, which is infinitely divergent for all times longer past than a definite determinable epoch. The distribution of heat at such an epoch is essentially initial that is to say, it cannot result from any previous condition of matter by natural processes. It is, then, well called an "arbitrary initial distribution of heat," in Fourier's great mathematical poem, because that which is rigorously expressed by the mathematical formula could only be realized by action of a power able to modify the laws of dead matter. In an article published about nineteen years ago in the Cambridge Mathematical Journal*, I gave the mathematical criterion for an essentially initial distribution; and in an inaugural essay, “De Motu Caloris per Terræ Corpus," read before the Faculty of the University of Glasgow in 1846, I suggested, as an application of these principles, that a perfectly complete geothermic survey would give us data for determining an initial epoch in the problem of terrestrial conduction. At the meeting of the British Association in Glasgow in 1855, I urged that special geothermic surveys should be made for the purpose of estimating absolute dates in geology, and I pointed out some cases, especially that of the salt-spring borings at Creuznach, in Rhenish Prussia, in which eruptions of basaltic rock seem to leave traces of their igneous origin in residual heatt. I hope this suggestion may yet be taken up, and may prove to some extent useful; but the disturbing influences affecting underground temperature, as Professor Phillips has well shown in a recent inaugural address to the Geological Society, are too great to allow us to expect any very precise or satisfactory results. (e.) The chief object of the present communication is to estimate from the known general increase of temperature in the earth downwards, the date of the first establishment of that consistentior status, which, according to Leibnitz's theory, is the initial date of all geological history. Feb. 1844.-"Note on Certain Points in the Theory of Heat," Mathematical and Physical Papers, by Sir W. Thomson, 1882, Vol. 1. Art. x. + See British Association Report of 1855 (Glasgow) Meeting. Much work in the direction suggested above has been already carried out by the Committee of the British Association, on Underground Temperature. temperature in earth's very im (f.) In all parts of the world in which the earth's crust has Increase of been examined, at sufficiently great depths to escape large in- downwards fluence of the irregular and of the annual variations of the super- crust; but ficial temperature, a gradually increasing temperature has been perfectly found in going deeper. The rate of augmentation (estimated at hitherto. only1th of a degree, Fahr., in some localities, and as much asth of a degree in other, per foot of descent) has not been observed in a sufficient number of places to establish any fair average estimate for the upper crust of the whole earth. But th is commonly accepted as a rough mean; or, in other words, it is assumed as a result of observation, that there is, on the whole, about 1° Fahr. of elevation of temperature per 50 British feet of descent. observed of heat out demon any present secular however (g.) The fact that the temperature increases with the depth Secular loss implies a continual loss of heat from the interior, by conduction of the earth outwards through or into the upper crust. Hence, since the strated: upper crust does not become hotter from year to year, there must be a secular loss of heat from the whole earth. It is possible that no cooling may result from this loss of heat, but only an exhaustion of potential energy, which in this case could scarcely be other than chemical affinity between substances but not so forming part of the earth's mass. But it is certain that either or past the earth is becoming on the whole cooler from age to age, or cooling, the heat conducted out is generated in the interior by temporary probable. dynamical (that is, in this case, chemical) action *. To suppose, as Lyell, adopting the chemical hypothesis, has done †, that the substances, combining together, may be again separated electrolytically by thermo-electric currents, due to the heat generated Fallacy of by their combination, and thus the chemical action and its heat electric perpetual continued in an endless cycle, violates the principles of natural motion. philosophy in exactly the same manner, and to the same degree, as to believe that a clock constructed with a self-winding movement may fulfil the expectations of its ingenious inventor by going for ever. * Another kind of dynamical action, capable of generating heat in the interior of the earth, is the friction which would impede tidal oscillations, if the earth were partially or wholly constituted of viscous matter. See a paper by Mr G. H. Darwin, "On problems connected with the tides of a viscous spheroid." Phil. Trans. Part 11. 1879. + Principles of Geology, chap. xxxi. ed. 1853. a thermo Exception to the soundness of arguments adduced in the promulgation and prosecution of the Huttonian reform. Secular diminution of whole amount of volcanic energy quite but not Chemical hypothesis to account for ordinary underground heat not impossible, but very impro bable. (h.) It must indeed be admitted that many geological writers of the "Uniformitarian" school, who in other respects have taken a profoundly philosophical view of their subject, have argued in a most fallacious manner against hypotheses of violent action in past ages. If they had contented themselves with showing that many existing appearances, although suggestive of extreme violence and sudden change, may have been brought about by long-continued action, or by paroxysms not more intense than some of which we have experience within the periods of human history, their position might have been unassailable; and certainly could not have been assailed except by a detailed discussion of their facts. It would be a very wonderful, but not an absolutely incredible result, that volcanic action has never been more violent on the whole than during the last two or three centuries; but it is as certain that there is now less volcanic energy in the whole earth than there was a thousand years ago, as it is that there is less gunpowder in a "Monitor" after she has been seen to discharge shot and shell, whether at a nearly equable rate or not, for five hours without receiving fresh sup plies, than there was at the beginning of the action. Yet this truth has been ignored or denied by many of the leading geologists of the present day*, because they believe that the facts within their province do not demonstrate greater violence in ancient changes of the earth's surface, or do demonstrate a nearly equable action in all periods. (i.) The chemical hypothesis to account for underground heat might be regarded as not improbable, if it was only in isolated localities that the temperature was found to increase with the depth; and, indeed, it can scarcely be doubted that chemical action exercises an appreciable influence (possibly negative, however) on the action of volcanoes; but that there is slow uniform "combustion," eremacausis, or chemical combination of any kind going on, at some great unknown depth under the surface everywhere, and creeping inwards gradually as the chemical affinities in layer after layer are successively saturated, seems extremely improbable, although it cannot be pronounced to be absolutely impossible, or contrary to all analogies in nature. The less * It must be borne in mind that this was written in 1862. The opposite statement concerning the beliefs of geologists would probably be now nearer the truth. |