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There is a marked difference between the results of this investigation and that deduced by Rowland* from the comparison of Joule's thermometers with his Baudin No. 6166. Tables VII. (referring to the first of the above series of comparisons) and VIII. (to the second) are intended to bring out this difference.

The numbers entered into the different columns of Table VII. are as follows:

Column I. TT or the reading on the Tonnelot thermo

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II. T-T; or the corresponding difference in the reading on the scale used by Joule and the Tonnelot scale.

III. T-TT, the correction to the Chappuis nitrogen-
scale as interpolated between the numbers
given in the table at the end of Guillaume's

IV. The calculated difference (T;-T) between the
Joule and Chappuis nitrogen-scale.

V. The corresponding difference (T-TN) R between
the Joule and Rowland's air-thermometers.
VI. The difference & between the numbers given in
Columns IV. and V.

A word of explanation is necessary as to how the numbers of Column V. have been obtained. Rowland gives in his paper the difference in the readings between Joule A and what he calls the "perfect" air-thermometer at a great number of points, none of them corresponding of course exactly to those of Column I., for which they are here required. I have taken the average between the value given for the temperature which lies nearest to that of Column I. and the two which lie immediately above and below it. The figures alter sufficiently slowly and with sufficient regularity to allow us to consider the numbers thus found as substantially correct.

We may deal more simply with the numbers obtained in the second series. The comparison between the Tonnelot and the Baudin thermometer already referred to gave, for the connexion between the two, the equation

Tr–Tr=0194–00089 Tr

Combining with this the experimental connexion between the Joule and Baudin,

TB-T0017+00084 TB,
T-T-0177-00005 TT;

*Proc. Amer. Acad. xvi. p. 38.

and from this we may calculate for the temperatures lying between 14° and 22° the difference between the Tonnelot and Joule readings. These are entered in Column II. of Table VIII.; the remaining columns have the same meaning as those in Table VII.

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The two series of measurements agree in showing a difference of nearly 0°05, which must be either due to a real difference between Rowland's "perfect" air-thermometer and that of Chappuis, or to some error in one or other of the comparisons.

We have no information at all as to how Joule proceeded in comparing together his thermometer with that of Rowland. The numbers furnished by Joule are obviously not those

obtained directly by experiment, as they are given to the thousandth part of a division. Joule probably only gave the mean between a certain number of successive observations ranging over several divisions of his scale. He would in this way eliminate the errors of division, and the regularity in the difference between his and Rowland's thermometer shows that some such process must have been adopted. There are certain corrections also no doubt applied by Joule, such as that due to the emergent stem, about which it would be necessary to have further information, before any definite conclusions can be drawn.

The important question as to a possible difference in the air-thermometers of Rowland and Chappuis can only be set at rest by a direct comparison of one of Rowland's thermometers with one compared at the Bureau International des Poids et Mesures.

But as regards the main point of the present investigation, this question does not arise. We are only concerned with Joule's thermometer, and the comparison between it and the Paris standard.

The relation between the intervals obtained by combining the two series of comparison was found to be, in terms of the Tonnelot nitrogen and hydrogen scales,

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Joule's final value for the equivalent of heat therefore reduces as follows :-

Joule's value for a temperature 61°-69 F. (16°5 C.).
On the scale of the French hard-glass thermometer.
On the scale of the nitrogen thermometer of the
Bureau International des Poids et Mesures
On the scale of the hydrogen thermometer of the
Bureau International des Poids et Mesures

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772.65 772-44



Rowland applies a small correction to Joule's value of the heat-capacity of his calorimeter. This would raise the equivalent by 2. Taking account of this, and considering that Joule's thermometer was never intended to measure temperatures nearer than one part in a thousand, and is not graduated sufficiently well to allow the decimal place to be determined with any certainty, we may state it as the result of this investigation that

Joule's equivalent of heat resulting from his own investigations

and reduced to the nitrogen thermometer of the Bureau International des Poids et Mesures is to the nearest unit 775 footpounds at the sea-level and the latitude of Greenwich. The number refers to a pound of water weighed in vacuo at a temperature of 61°7 F. (16°5 C.).

The equivalent reduced to ergs becomes 4·173 × 10−7.

It is not necessary to discuss the older observations of Joule, or to modify his numbers by attaching weights to his experiments different from those which he gave to them himself. The result of Joule's last paper, as reduced by himself, should be taken as his final judgment. Rowland's value at 16°5 is 4.186 x 107, but the results of this paper open out the possibility that this number might have to be reduced somewhat when referred to the Paris air-thermometer. It seems most probable that the correct value of the equivalent lies somewhere between Joule's value and that of Rowland. The higher values obtained by Mr. Griffiths and myself and Gannon by the electrical method are not easily accounted for, but for the present they cannot in my opinion be put into competition with the direct determinations of Joule and Rowland. The dis

crepancy no doubt will be cleared up. In the meantime a comparison between one of Rowland's thermometers and the Paris standard would be of great interest.

XLIX. On the Kinetic Energy of the Motion of Heat and the corresponding Dissipation Function. By Dr. LADISLAS NATANSON, Professor of Natural Philosophy, University of Cracow*.

1. N the following the fundamental assumptions of the former paper ("On the Kinetic Interpretation of the Dissipation Function") will be adopted. A fluid medium is considered which is supposed to consist of a multitude of moving molecules. Let u, v, w be the components of the "molar" velocity, i. e. of the mean velocity of the molecules within an element de dy dz; and let , n, be the components of the individual velocity of any given molecule in that element. We will employ the symbol p to denote the density of the medium; and Q to denote any property of a molecule which depends on the values of (u+¿), (v+n), and (w+5). Let Q indicate the mean value of Q for all molecules within

*Translated from "Rozprawy" (Transactions) of the Cracow Academy of Sciences, Math. and Phys. Section, vol. xxvii. Communicated by the Author,

an element, and X, Y, Z the components of acceleration due to external forces at the point (x, y, z). We will write dQ/dt the total or actual variation of Q; and by &Q/dt we will represent such variation of Q as can be due to the mutual interference between molecules. Then (Maxwell, "On the Dynamical Theory of Gases," Scientific Papers, ii. p. 26),

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If we give to Q in this equation consecutively such significations as are consistent with definition and then eliminate terms including X, Y, Z and 8/dt, a set of propositions will be obtained, constituting what may be called a Kinematical Theory of Fluids, a theory of rather high degree of generality which must not be confounded with special molecular theories of usually very hypothetical character. It is with Hydrodynamics that the kinematical theory seems to be most intimately connected, the fundamental hydrodynamical equations (or possibly some generalizations thereof) being simple deductions from the equations of that theory.

2. In equation (1) put


Q= (u + §) { (u + §) 2 + (v + n)2 + (w +5)2}; . . (2)

and let us write for brevity,

§ (§2+n2+52)=1°2•

Neglecting small terms we obtain



dt {u (u2 +v2+w2) +u(352+n2+52)+rz} .


+ · {p¿2(3u2+v2+w2) +p§2(§2+n2+52)}



+ (Quvpn3) + 2 = (2uwp513) = 2pu


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+pX(3u2+v2+w2 +3 §2 +n2 + 52)+2pYuv+2pZuw. (4)

We shall simplify this equation, being satisfied with a first approximation. Put in (1) Q= (u+)2; put, again, Q=u+§, multiply by 2u, and to a first approximation it follows,

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