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sion another is immediately deduced, which is of fundamental importance in connexion with the question now under consideration. It follows that the shell and nucleus must rotate together very nearly as if one mass, for otherwise, according to a result obtained by Mr. Hopkins*, the annual precession of the equinoxes would greatly differ from that which is observed. Hence it follows that great friction and pressure must exist at the surface of contact of the nucleus and shell, and probably also even between the particles of the fluid. Such a state of things should indeed be expected à priori from the highly crystalline structure, and probably unequal surface of the shell, as well as the viscous nature of the fused materials of the earth, so far as we can judge from those coming under our noticet. The existence of great pressure has been already indicated in connexion with the gradual tendency of the nucleus to expand its volume.

While the strata of equal density in the shell would thus increase in oblateness from the outer to the inner surface, the strata of equal pressure in the nucleus would always follow the opposite law, of decreasing in oblateness from the surface to the centre. I have shown that this result, combined with that which has been just mentioned, would lead to another directly connected with the question of the earth's rotation, namely that the difference between the greatest and least moments of inertia of the earth continually tends to increase during the process of solidification, and consequently that the stability of the earth's axis of rotation, so far from being disturbed by that process, is increased during its successive stages. This point was further developed in a letter to Sir John Lubbock ‡, replying to some communications with which he had favoured me with reference to a short paper he had inserted in the Journal of the Geological Society of London, wherein he had endeavoured to show the possibility of a change in the position of the earth's axis of rotation from the effect of physical changes in its structure.

The increase of the difference between the greatest and least moments of inertia of the earth would be chiefly due to an absolute increase of the former. As this corresponds to the present axis of rotation, it follows, that not only would the stability of the earth's axis of rotation be more completely assured, but also that its velocity of rotation would be diminished. This conclusion, combined with that already arrived at from a general consideration of the influence of the process of solidification, seems to establish the existence of a tendency to increase the length of the day. At the same time the slow cubical contrac

* Phil. Trans. 1840, p. 207; see also Phil. Trans. 1851, p. 546.
+ See Berzelius in Poggendorff's Annalen, vol. xliii.

1 Proceedings of the Royal Society, February 1852.

tion of the entire spheroid, from the gradual cooling of all its particles, would lessen its dimensions, and thus tend to accelerate the velocity of rotation so as to diminish the length of the day. The energy of both of these opposing tendencies depends upon a common cause, the rate of cooling of the entire earth. This has been demonstrated by Fourier and other illustrious mathematicians, on the most favourable suppositions to rapidity of refrigeration, to be so extremely slow, that if only one of the counteracting influences here adduced existed without the other, we could scarcely expect to discover its action on the rotation of the earth until after a long period of exact observations. When the fact of the simultaneous existence and opposition of these influences is remembered, it should not excite surprise that astronomical observations should have hitherto never disclosed any variation in the length of the day, and ages may possibly elapse before any such variation will be discovered.

XII. On Heat as the Equivalent of Work.

By W. J. MACQUORN RANKINE, C.E., F.R.SS.L. & E. To the Editors of the Philosophical Magazine and Journal. GENTLEMEN,


N the Number of your Magazine for July (at p. 75) there appears an abstract of a paper by M. Hoppe, first published in Poggendorff's Annalen, vol. xcvii. p. 30, commencing, "Hoppe has contributed a memoir upon this most interesting and important subject, which places the analytical theory in a remarkably clear, simple, and general point of view, so far at least as it relates to permanent gases." This observation naturally leads the reader to infer, that the theory previous to the publication of M. Hoppe's paper was deficient in clearness, simplicity, and generality; and on first reading it in Silliman's Journal, I contemplated entering into a detailed discussion of M. Hoppe's paper; but this intention I have since abandoned, on finding that that discussion has been already made by Professor Clausius (Poggendorff's Annalen, vol. xcvii. p. 173) in a short paper, which appears to me to be well worthy of publication in the English language. I shall therefore confine my remarks to stating, that the whole theory of heat as the equivalent of work performed by the expansion of elastic substances, whether gaseous, liquid, or solid, is summed up in this one equation:

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where J is Joule's equivalent; dq the quantity of heat received by the substance during the increase of absolute temperature dr


and of volume du; k the real specific heat of the substance p the pressure under which it expands, so that pdv is the external work performed by the expansion. The various expressions for this equation which have appeared in the writings of Messrs. Clausius and Thomson, and in my own, merely differ in form, and are all substantially equivalent to each other. It affords the solution of every conceivable question where the mutual relations of heat and of work by cubic expansion are concerned, and has been abundantly and rigorously verified by experiment; and I think I am justified in maintaining it to be clear and simple as well as general.

I trust it will be understood, that, in making this statement, I have no wish to detract from the merits of M. Hoppe, my sole object being to defend the existing theory against the prejudicial inference which might be drawn from the opening remark of the English abstract of M. Hoppe's paper-a remark which does not appear in the original German.

I have the honour to be, Gentlemen,

Glasgow, July 2, 1856.

Your most obedient Servant,


P.S. I may take this opportunity of giving increased publicity to a peculiar transformation of the equation A, which is useful in certain special investigations. The paper in which it was first given was read to the Royal Society of Edinburgh in February 1855, but has not yet been published.

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Povo, To denote the pressure, volume, and absolute temperature of the substance in the ideal state of perfect gas.-W. J. M. R.

XIII. On the Demonstration of Fresnel's Formulas for Reflected and Refracted Light; and their Applications.-Part II. By the Rev. BADEN POWELL, M.A., F.R.S. &c., Savilian Professor of Geometry in the University of Oxford*.

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I N a former paper (see Phil. Mag. &c. July 1856) I have placed in a connected point of view the several principles and deductions leading to the well-known formulas of Fresnel, as well as to certain modifications of them, for the amplitudes of the vibrations of the incident, reflected, and refracted rays, whether polarized parallel or perpendicular to the plane of incidence. I have also remarked on the question which has so long

*Communicated by the Author,

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divided opinions, whether the vibrations are parallel or perpen-. dicular to the plane of polarization, and on the decisive evidence lately obtained in favour of the latter hypothesis. Some other questions relative to the same subject still demand examination, to which I propose now to refer.

2. For this purpose it will be necessary briefly to premise a recapitulation of the primary principles on which the several investigations proceed, and which are fully discussed in my former


These are,

I. The principle of vis viva; (1) that the square of the velocity multiplied by the vibrating mass is the true measure of force; (2) that the vis viva of the incident vibrations is equal to the sum of the vires viva of the reflected and refracted vibrations. Or, m and m, being the simultaneously vibrating masses of æther without and within the medium, h, h', h, respectively the amplitudes (which are the measures of the velocities) of the incident, reflected, and refracted vibrations ;-then the law of vis viva is expressed by the equation

m(h2 — h12)=m,h2.

II. The law of equivalent vibrations, which on Maccullagh's view is expressed by

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(a) h+h=h, for vibrations perpendicular to the plane of incidence,

and (i and r being the angles of incidence and refraction)


(B) h+h=h; for vibrations parallel to the plane

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of incidence.

III. On the principle adopted by Fresnel in the second case (B), the same law would be expressed by

(a)... h—l'=h, for vibrations perpendicular to the plane of incidence,

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COS r cos i

h-h'=h for vibrations parallel to the plane

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VI. Maccullagh's hypothesis of vibrations parallel to the plane

of polarization.

VII. Fresnel's hypothesis of vibrations perpendicular to the plane of polarization.

3. By different combinations of these principles, different modifications of the formulas result. Thus we have the hypotheses

(A) combining Nos. I. II. IV. VI., whence are obtained the formulas

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+tan (i+r)

; — ....k, = (1 +

tan (i-r)


+tan (+7) ... perpendicular,

which are Maccullagh's formulas; the double sign indicating the change at the polarizing angle.

(B) Combining Nos. I. II. V. VII., whence are obtained,

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+tan (i+r)

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tan (ir) cos i
+tan (i+r)


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(C) Combining Nos. I. III. V. VII., whence are obtained,

2 sin r cos i
sin (i+r)

tan (i+r)

+tan (i+r) cos r

4. Each of these two last sets differs from Fresnel's in the signs. Fresnel's original formulas can only be produced from assuming hypothesis (B) for h, and (C) for k; or we haye,— (D) combining Nos. I. IIa. IIIß. V. and VII., whence are obtained,


-sin (i-r)
sin (i+r)

2 sin r cos i


sin (i+r)

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5. With regard to the law of equivalent vibrations, it may indeed be remarked that Prof. Maccullagh in stating it, with a view to his ulterior researches on crystalline reflexion, rather assumes than demonstrates the main principle, and thus the modified form of that law (No. III.) may possibly be as open to consideration as the original form. But as neither form exclusively will produce Fresnel's original formulas, it becomes of more importance to look to some other principle which might

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