sphere, will keep the top of the tube sufficiently moist for an immense time.

The only possible changes which can go on in this cell are in the zinc and the solution in immediate contact with it. This solution can at any time be drawn off with a pipette and replaced by fresh, without greatly affecting the liquid in the bottle (if the cork be air-tight); and the zinc can still more easily be taken out and replaced by a new piece.

I have described the cell as at present made; but if there were any chance of its coming into use as a standard, a few modifications might be introduced. Thus the zinc might be a short rod with an india-rubber collar fitting the tube and with a short copper wire attached to it, which should project above the cork instead of the zinc, the joint being a little way down the tube and protected by a coat of varnish from damp air. A set of experiments would have to be made to determine the dependence of electromotive force on temperature; and then a thermometer with a short scale might be fixed in each cork.

University College, London.

II. Cr the Thermoelastic, Thermomagnetic, and Pyroelectric Properties of Matter. By WILLIAM THOMSON, M.A., late Fellow of St. Peter's College, Cambridge, Professor of Natɩral Philosophy in the University of Glasgow*.

1. A

BODY which is either emitting heat, or altering its dimensions against resisting forces, is doing work upon matter external to it. The mechanical effect of this work in * This paper is in the main a reprint from an article which appeared under the title "On the Thermoelastic and Thermomagnetic Properties of Matter, Part I.," in April 1855, in the first number of the " "Quarterly Journal of Mathematics,' but which was confined to the thermoelastic part of the subject. The continuation, in which it was intended to make a similar application of thermodynamic principles to magnetic induction, was never published or written; but the results which it should have contained were sufficiently indicated in a short article on "Thermomagnetism," which I wrote at the request of my friend and colleague the late Professor J. P. Nichol for the second edition of his 'Cyclopædia,' published in 1860, and which I include in the present reprint. The addition of "Pyro-Electricity," which I now make to the title of the former article, is justified by another short quotation from the second edition of Nichol's Cyclopædia' (article "Thermo-Electricity, Division I.--Pyro-Electricity, or Thermo-Electricity of Nonconducting Crystals"), and a short addition, now written and published for the first time, in which the same thermodynamic principles are applied to this form of thermoelectric action.

Several additions both in the shape of text and footnote are appended in the course of the reprint. These are all distinguished by being enclosed in brackets, [ ].]

one case is the excitation of thermal motions, and in the other the overcoming of resistances. The body must itself be altering in its circumstances, so as to contain a less store of energy within it, by an amount precisely equal to the aggregate value of the mechanical effects produced; and conversely, the aggregate value of the mechanical effects produced must depend solely on the initial and final states of the body, and is therefore the same, whatever be the intermediate states through which the body passes, provided the initial and final states be

the same.

2. The total intrinsic energy of a body might be defined as the mechanical value of all the effect it would produce, in heat emitted and in resistances overcome, if it were cooled to the utmost, and allowed to contract indefinitely or to expand indefinitely according as the forces between its particles are attractive or repulsive, when the thermal motions within it are all stopped; but in our present state of ignorance regarding perfect cold, and the nature of molecular forces, we cannot determine this "total intrinsic energy" for any portion of matter; nor even can we be sure that it is not infinitely great for a finite portion of matter. Hence it is convenient to choose a certain state as standard for the body under consideration, and to use the unqualified term intrinsic energy with reference to this standard state; so that the "intrinsic energy of a body in a given state" will denote the mechanical value of the effects the body would produce in passing from the state in which it is given, to the standard state-or, which is the same, the mechanical value of the whole agency that would be required to bring the body from the standard state to the state in which it is given.

3. In Part V.* of a series of papers on the Dynamical Theory of Heat, communicated to the Royal Society of Edinburgh, a system of formulæ founded on propositions established in Part I.† of the same series of papers, and expressing, for a given fluid mass, relations between its pressure, its thermal capacities, its intrinsic energy (all considered as functions of its temperature and volume), and Carnot's function of the temperature, were brought forward for the purpose of pointing out the importance of making the intrinsic energy of a fluid in different states an object of research along with the other elements which have hitherto been considered, and partially investigated in some cases. In the present communication a similar mode of treatment, extended to include solid bodies, unmagnetic [and unelectrified], or magnetized [or electrified] * Trans. Roy. Soc. Edinb. December 15, 1851.

+ Ibid. March 17, 1851.

in any way, is shown to lead to the most general possible theory of elasticity, whether of solids or fluids, and to point out various thermodynamic properties of solids and various thermal effects of magnetism [and of electricity] not hitherto discovered.

SECTION I.-Elasticity of Solids or Fluids not subjected to

Magnetic Force.

4. Let x, y, z, E, n, Y be six independent variables expressing the mechanical condition of a homogeneous solid mass, homogeneously strained in any way, and let t be its temperature; and (in accordance with the preceding explanations) let e denote its intrinsic energy, reckoned from a certain "standard state" defined by particular values, xo, yo, 20, o, no, So, to, on which its physical condition depends. Thus, if denotes a certain function depending on the nature of the substance, and vanishing for the values ao, Yo,... to of the independent variables, we have

(1)

e=p(x, y, z, §, n, y, t) ; and a knowledge of the function [with besides a knowledge of w for one particular temperature†] comprehends all the thermoelastic qualities of the solid.

5. Now let us suppose the body to be strained so as to pass from the mechanical state (co, yo, zo, ko, no, So) to (x, y, z, §, n, 5) while it is constantly kept at the temperature t; and let H denote the quantity of heat that must be supplied to it during this process to prevent its temperature from being lowered (a quantity which of course is zero, or negative, for such strains as cause no thermal effects, or which cause positive evolutions of heat). Let the body be brought back to its mechanical condition (xo, yo, zo, o, no, So) through the same or any other of all the infinitely varied successions of states by which it may be made to pass from one to the other of the two which have been named, its temperature being kept always at t. Then, by the second Fundamental Law of the Dynamical Theory of Heat (see Trans. Roy. Soc. Edinb. May 1, 1854, p. 126), we must have H H' +

and therefore H'=-H.

=0,

* The terms a strain, or to strain, are used simply with reference to alterations of dimensions or form in a solid-the forces by which " a strain" is produced being called the straining tensions or pressures, or sometimes merely the tensions or pressures, to which the solid is subjected. This distinction of terms is adopted in accordance with the expressions used by Mr. Rankine in his paper on the Elasticity of Solids (Cambridge and Dublin Mathematical Journal, February 1851).

† [See equations (10), (11) of § 7 below.]

6. We conclude that the quantity of heat absorbed by the body in being strained from one state to another at the same temperature is quite independent of the particular succession of states through which it is made to pass, provided it has throughout the same temperature. Hence we must have

(2)

H=¥(x, y, z, §, n, y, t) —¥(xo, Yo, zo, Eo, no, So, t), where denotes a function of the variables. Now the mechanical value of the heat taken in by the body while it passes from one condition to the other, together with the work spent in compelling it to do so, constitutes the whole augmentation of mechanical energy which it experiences; so that if e denote this augmentation—that is, if

e=p(x, y, z, §, n, 5, t) — p(xo, Yo, zo, ko, no, sv, t), (3) and if u denote the work done by the applied forces and J the mechanical equivalent of the thermal unit, we have

From this we conclude that the work required to strain the body from one to another of two given mechanical states, keeping it always at the same temperature, is independent of the particular succession of mechanical states through which it is made to pass, and is always the same when the initial and final states are the same. This theorem was, I believe, first given by Green (as a consequence of the most general conceivable hypothesis that could be framed to explain the mutual actions of the different parts of a body on which its elasticity depends), who inferred from it that there cannot be 36, but only 21, independent coefficients [or "moduluses"] of elasticity, with reference to axes chosen arbitrarily in any solid whatever. It is now demonstrated as a particular consequence of the Second General Thermodynamic Law. It might at first sight be regarded as simply a consequence of the general principle of mechanical effect; but this would be a mistake, fallen into from forgetting that heat is in general evolved or absorbed when a solid is strained in any way; and the only absurdity to which a denial of the proposition could lead would be the possibility of a self-acting machine going on continually drawing heat from a body surrounded by others at a higher temperature, without the assistance of any at a lower temperature, and performing an equivalent of mechanical work.

7. The full expression of the Second Thermodynamic Law for the circumstances of elastic force is, as is shown in the passage referred to above (Trans. Roy. Soc. Edinb. May 1, 1854, p. 126), that if H, H't, &c. denote the quantities of heat emitted

from a body when at temperatures* t, t' respectively, during operations changing its physical state in any way, the sum

H

t

Σ must vanish for any cycle of changes, if each is of a perfectly reversible character, and if at the end of all the body is brought back to its primitive state in every respect. Let us consider, for instance, the following cycle, which obviously fulfils both conditions.

(I.) Let the body, initially in the state (xo, Yo, zo, §o, no, So, t), be raised in temperature from t to t', its form and dimensions being maintained constant.

(II.) Let it be strained from the state (xo, yo, zo, o, No, So) to the state (x, y, z, §, n, ), while its temperature is kept always at t'.

(III.) Let it be lowered in temperature from t' to t, its form and dimensions being retained.

(IV.) Let it be brought back to the mechanical state (To, yo, 20, Eo, no, ), while its temperature is kept constantly

at t.

The quantities of heat taken in by the body in these successive operations are respectively :

1

(I.) {P(xo, Yo, zo, §o, No, Co, t') — 4(xo, Yo, zo, §o, No, So, t)},

hecause the difference of the whole mechanical energies is simply the mechanical value of the heat taken in or emitted in all cases in which no work is either done on the body or received by it in virtue of the action of applied forces;

(II.) ↓(x, y, z, §, n, Y, t') —↓(xo, Yo, zo, §o, no, So, t'), according to the notation expressed by equation (2) above ;

(III.) − } {$(x, y, z, §, n, 5, t') −p(x, y, z, §, n, 5, t)},

and

(IV.) —{†(x, y, z, §, n, y, t) —¥(xo, yo, 20, §o, no, so, t)}.

• Reckoned on the absolute thermodynamic scale, according to which "temperature" is defined as the mechanical equivalent of the thermal unit divided by "Carnot's function." In a paper "On the Thermal Effects of Fluids in Motion," by Mr. Joule and myself, communicated to the Royal Society last June [1854], and since published in the Philosophical Transactions, it is shown that temperature on the absolute thermodynamic scale does not differ sensibly from temperature on the ordinary scale of the air-thermometer, except by the addition of a constant number, which we find to be about 273-7 for the Centigrade scale. Thus, on the system now adopted, the temperature of melting ice is 273-7, that of boiling water is 373.7, and differences of temperature are sensibly the same as on an ordinary standard Centigrade thermometer.