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A transformation which thus remains at the conclusion of a circular process without another opposite one, and which according to this theorem can only be positive, we shall, for brevity, call an uncompensated transformation.

The different kinds of operations giving rise to uncompensated transformations are, as far as external appearances are concerned, rather numerous, even though they may not differ very essentially. One of the most frequently occurring examples is that of the transmission of heat by mere conduction, when two bodies of different temperatures are brought into immediate contact; other cases are the production of heat by friction, and by an electric current when overcoming the resistance due to imperfect conductibility, together with all cases where a force, in doing mechanical work, has not to overcome an equal resistance, and therefore produces a perceptible external motion, with more or less velocity, the vis viva of which afterwards passes into heat. An instance of the last kind may be seen when a vessel filled with air is suddenly connected with an empty one; a portion of air is then propelled with great velocity into the empty vessel and again comes to rest there. It is well known that in this case just as much heat is present in the whole mass of air after expansion as before, even if differences have arisen in the several parts, and therefore there is no heat permanently converted into work. On the other hand, however, the air cannot again be compressed into its former volume without a simultaneous conversion of work into heat.

The principle according to which the equivalence-values of the uncompensated transformations thus produced are to be determined, is evident from what has gone before, and I will not here enter further into the treatment of particular cases.

In conclusion, we must direct our attention to the function T, which hitherto has been left quite undetermined; we shall not be able to determine it entirely without hypothesis, but by means of a very probable hypothesis it will be possible so to do. I refer to an accessory assumption already made in my former memoir, to the effect that a permanent gas, when it expands at a constant temperature, absorbs only so much heat as is consumed by the external work thereby produced. This assumption has been verified by the later experiments of Regnault, and in all probability is accurate for all gases to the same degree as Mariotte and Gay-Lussac's law, so that for an ideal gas, for which the latter law is perfectly accurate, the above assumption will also be perfectly accurate.

The external work done by a gas during an expansion dv, provided it has to overcome a pressure equivalent to its total expansive force p, is equal to pdv, and the quantity of heat absorbed Phil. Mag. S. 4. Vol. 12. No. 77. Aug. 1856.

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and by substituting this value of in the equation (13), the

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But, according to Mariotte and Gay-Lussac's law,

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where a is the inverse value of the coefficient of expansion of the permanent gas, and nearly 273, if the temperature be given in degrees C. above the freezing-point. Eliminating p from (15) by means of this equation, we have

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It is of no importance what value we give to this constant, because by changing it we change all equivalence-values proportionally, so that the equivalences before existing will not be disturbed thereby. Let us take the simplest value, therefore, which is unity, and we obtain

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According to this, T is nothing more than the temperature counted from a°, or about 273° C. below the freezing-point, and, considering the point thus determined as the absolute zero of temperature, T is simply the absolute temperature. For this reason I introduced, at the commencement, the symbol T for the reciprocal value of the function f(t). By this means all changes which would otherwise have had to be introduced in the form of equations, after the determination of the function, are rendered unnecessary; and now, according as we feel disposed to grant the sufficient probability of the foregoing assumption or not, we may consider T as the absolute temperature, or as a yet undetermined function of the temperature. I am inclined to believe, however, that the first may be done with hesitation.

XI. On the Influence of the Earth's Internal Structure on the Length of the Day. By HENRY HENNESSY, M.R.I.A., Professor of Natural Philosophy in the Catholic University of Ireland*.

THE HE period of a complete revolution of the earth around its axis of rotation, depends not only on the dimensions of the mass, but also on the distribution of the particles of which it is composed. The variations which in the course of ages may possibly take place in the distribution of these particles will therefore tend to produce some change in the length of the day. The question of the secular variation of the earth's velocity of rotation has been already, to a certain extent, examined by Laplace+; but his investigation has regard solely to the contraction of the dimensions of the globe considered as a slowly cooling solid. If the interior of the earth is in a state of fluidity from heat, or in other words, if it consists of a solid exterior shell filled with a nucleus of matter in a state of fusion, from which state the portion now solid had gradually passed to its present condition, the inquiry assumes a different shape. Hitherto this, as well as every other question connected with the general structure and rotation of the earth, has been treated on the assumption that the portions composing the fluid underwent no change in their positions on entering into the solid state. This assumption formed not only the basis of the inquiries of Laplace, but even of more recent investigators, who professed to consider the purely physical conditions of problems relating to the structure of our planet in a far more complete manner. The change of state of the matter composing the interior of the earth in passing from fluidity to solidity seems to have been thus entirely overlooked, although a little reflection might have suggested that such a change would possibly influence its structure to a very considerable extent.

In the first part of my "Researches in Terrestrial Physicst," the necessity of attending to this circumstance was distinctly pointed out, and the superfluous nature of the contrary assumption was formally declared. Experiments were quoted in the second part confirmatory of my views on this point, which I believe have never since been called in question. When we reflect on the remarkable results of the researches of Professor Bischof, M. Ch. Sainte-Claire Deville, and M. Delesse, upon the contraction of the principal materials of the solidified crust of the earth in passing successively from the fluid state through

* Communicated by the Author.

† Mécanique Céleste, livre xi.

Philosophical Transactions, 1851, part 2, p. 495.

the vitreous condition up to that of complete crystalline structure, it is at once apparent that the influence of changes of the state of molecular aggregation of such masses must exercise an important influence on the general structure of the earth, as well as the slow cubical contraction of the entire mass during the process of its refrigeration.

In my second memoir, I showed the way in which the former species of contraction would affect the interual structure of the earth supposed to consist of a solid shell and included nucleus of fluid. The process of solidification of the fluid, commencing from the centre and ending at the surface, according to the views of Poisson, was proved to be impossible; and the only way in which the process could take place, was shown to be by successive additions of matter from the surface of the nucleus to the inner surface of the solid shell. Each outer stratum of the nucleus would thus successively become a stratum of the shell. But the density of any stratum of equal pressure in the fluid depends on the pressure of all the strata by which it is enveloped, and therefore the removal of these strata in regular succession, gradually tending to decrease the pressure, must decrease the density of the stratum in question. This action, operating on all the strata of the nucleus, will manifestly give it a tendency to enlarge its volume so as to fill up the space left by the contraction of its exterior strata. This expansion of the nucleus will evidently be accompanied by a diminution of its mean density, and has been shown to be also attended with a change in the law of density in going from its centre to its surface. The density will vary less rapidly as the solidification of the mass advances, and the fluid will tend to become more homogeneous. The mass of the shell is at the same time continually augmented in a corresponding degree by the successive additions it thus receives at its inner surface, so that the aggregate effect of the process of solidification is the removal of matter from the centre towards the surface of the earth.

An admirable example of the effect of internal forces on the solidification of a cooling mass of fused igneous rock, which will serve in some measure to illustrate my remarks, has been described by Mr. Darwin in his 'Naturalists' Voyage*;' it presents a case where the forces tending to expand the liquid enclosed in the first solidified envelope are extremely energetic, and the order of the phænomena is thus very clearly exhibited. He noticed in several places in the Island of Ascension volcanic bombs, which have been shot through the air while in a fluid state, and which, spinning around their centres as they passed through their course, have usually assumed a rounded shape. He then remarks, that * Page 493.

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not only the external form, but in several cases their internal structure, shows that these bodies have revolved in their aërial course. One of these bombs when broken through presented the following appearances:-1. The central part was highly cellular, the cells decreasing in size from the centre towards the exterior. 2. These cells terminated at a shell-like case of compact stone about a third of an inch in thickness. 3. Outside this case was a crust of finely cellular lava. Mr. Darwin explains these phænomena very rationally, by saying that the outermost crust cooled rapidly into the state in which it came under his observation; then the enclosed fluid was pressed against the shell so formed by centrifugal force arising from the rapid rotation of the mass, thus producing the compact stony casing; lastly, the action of the same force relieving the pressure of the fluid at its central portions, the expansive tendency of the included elastic vapours would ultimately produce the coarse cellular structure at the centre. It might be added, that the passage of the fluid into the solid state, when forming the compact case of stone, being necessarily accompanied by contraction, would allow some space for the operation of the expansion of the remaining fluid.

If the removal of matter from the interior towards the exterior portions of the earth took place equally along every radius drawn from the centre to the surface, all of the earth's moments of inertia would be augmented, and the length of the day, so far as it could be affected by this cause, would of course be increased. The contraction of the solidifying surface of the nucleus upon the inner surface of the shell, in passing from the fluid to the solid state, taking place from within outwards, its effect could not be that suggested by M. Delesse*, in terminating his valuable remarks on the crystalline contraction of rocks, namely a diminution of the earth's radius, and a consequent increase in the velocity of rotation, but precisely the reverse.

On the old assumption of mathematical investigators, that the particles composing the earth retained the same positions on passing from the fluid to the solid state, the oblateness of the strata of equal density of the solidified mass would correspond to the surfaces of equal pressure in the fluid, and would be less and less oblate in going from the outer to the inner surface of the shell. By discarding this assumption, and admitting the influence of the physical changes that take place in the passage of the fluid matter of the nucleus to the state of solidity, I have shown that the process of solidification already briefly described would tend to augment the oblateness of the strata of the shell in going from its outer to its inner surface. From this conclu* Comptes Rendus, vol. xxv. p. 545.

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