not the least interesting of these is the problem of the three states of matter and the difference in their energy contents. Following on the theory of allotropy, it is an obvious conclusion to draw that the three states are equilibrium mixtures of different molecular phases. Whilst it cannot be assumed that any change of state is due to a change of phase of every molecule present, yet it is not impossible that, at any rate, the latent heat of evaporation should approximate to the amount of energy associated with a change of phase-that is to say, one or more quanta at the infra-red fundamental for every molecule. The values of the infra-red fundamentals are at present known for very few substances, but it is possible to make use of the most important molecular frequency, in the infra-red, which, as I have shown, is the infra-red fundamental in the cases of water and sulphur dioxide. In Table I. are given the relations between the latent heat of evaporation per molecule and the quantum at the infra-red frequency. The agreement is so close that I have calculated the latent heat on the basis of it being equal to the number of infra-red quanta shown in the seventh column. The calculated values expressed as calories per gram-molecule are given in the last column.

In the cases of carbon tetrachloride and sulphur it may be noted that the infra-red measurements are not so trustworthy as those of the other compounds *. At the same time it must be noted that there are some compounds for which this relation does not appear to hold. Now, it is well known that water is associated in the liquid condition, and therefore the suggestion may be made that when the latent heat is equal to two or more infra-red quanta, this is due to the molecule of the liquid being more complex than the molecule of the vapour. The cases in which the relation does not hold would then be explicable on the ground that the molecular association is indefinite-that is to say, the liquid contains an equilibrium mixture of two molecular complexes. This application of the molecular phase hypothesis cannot, however, be further discussed here.

In conclusion, reference may be made to the deduction from the molecular phase hypothesis that, since a specific molecular phase is essential for a molecule to take part in a given reaction, a definite amount of energy must, in general, be supplied to the molecule before the reaction can take place. Whilst this is not new, the conception of the critical

* Coblentz, Pub. Carnegie Inst. Washington, No. 35 (1905).

increment of energy being due to Marcelin and Rice*, the present theory independently arrives at the same conclusion, and, moreover, shows that the critical increment is equal to a whole number of quanta at the infra-red fundamental. In a series of papers, Lewis † has made certain calculations of the critical increments necessary in certain reactions and the frequencies at which this energy is absorbed. He writes the expression given above in the form

Observed heat of reaction = E-E1,

where E, is the sum of the critical increments of the reactant molecules and E, that of the resultant molecules. When the energy is absorbed or evolved at the phase frequency, this expression will only be true if the reactant molecules are completely dissociated into atoms. It is evident from the present hypothesis that the critical increment is a whole number of quanta at the infra-red fundamental-an amount of energy which may be very much smaller than one quantum at the phase frequency. In such a case impossible results will be obtained if the frequencies are calculated from the critical increments derived from the heats of dissociation of the molecules into atoms.

A

RECENT paper § has broken ground for a rectified comparison between Einstein's kinematics and Newton's methods by insisting first upon restoring the broader connexion of force with "Quantity of motion." New tribute is paid to the hypnotic power of a current view to evade challenge, in the delay of this conclusive step towards juster correlation, after the disadvantage within the range of electromagnetics had been conceded against the narrowed form of Newton's equations. But an undeniable habit has inclined overmuch toward some particulars of rigid dynamics, however aware one is latently that the literal condition of variable mass is often directly natural, and that the devices lie close at hand which fit for parallel treatment the enlarged (or

* Rep. Brit. Assoc. p. 397 (1915).

†Trans. Chem. Soc. 1914 to 1919, passim.
Communicated by the Author.

§ Slate, Phil. Mag., April 1920, p. 433.

more figurative) concept of inertia *. Else why Minkowski's words?" Die Gleichungen der Newtonschen Mechanik zeigen. eine zweifache Invarianz. Einmal bleibt ihre Form unverändert wenn man das zugrunde gelegte räumliche Koordinatensystem einer beliebigen Lagenveränderung unterwirft, zweitens, wenn man es in seinem Bewegungszustande verändert, nämlich ihm irgendeine gleichförmige Translation aufprägt"†. In the origins of relativity, the equivalent of equations (1, 2, 3) below seems nowhere drawn into consideration. Einstein and Lorentz do not break away radically from the former's equation:

"Massenzahl × Beschleunigungszahl = Kraftzahl".

This type of relation is in evidence unaltered, even where a variable "transverse mass enters. For the Newtonian procedure, a force thus expressed implies an inertia held stationary at its epoch-value; the derivative of momentum is partial. We shall meet an important instance presently §.

The more thoroughgoing ascription of variable inertia to electrons, as a starting-point either for Newtonian dynamics or for relativity, carries with it then nothing forced, nor subtle, nor unique into the equations of motion, activity, and work. Grant freely how much a dense ignorance about detail in atomic and electronic systems compels us to leave vague, beyond the proof of inertia through observed partitions. of energy. But neither Newton nor Einstein escapes that

* Careful elementary comment upon Newton's laws will not neglect this side. See Tait and Steele (1878), Dynamics of a Particle,' p. 330, etc.; Slate (1900), 'Principles of Mechanics,' p. 220, etc. Passing over to moment of inertia, it may change with the rotation-axis. The idea of "effective inertia," expanded fruitfully by Sir J. J. Thomson, includes variableness of that quantity, especially where undetected (or ignored) forces are in play. This latter phase, in its application to electrons, I have indicated earlier: 'Science1 (1908), vol. xxviii. p. 180.

† Raum und Zeit. Conveniently accessible in the collection of foundation-laying papers (cited here as Sammlung"): Lorentz-Einstein Minkowski; Das Relativitätsprinzip, Teubner (1913). p. 56; and p. 72.

See

Sammlung, p. 51. See also (p. 28): "Die zu entwickelnde Theorie stützt sich .... auf die Kinematik des starren Körpers." The position taken by Lorentz shows on pp. 17, 78, etc. Expository writers like Silberstein conform : see his' Theory of Relativity,' pp. 15, 17; Chapter VII, pussim; etc. This is an eminently temperate summary. The terms of all such statements are at best misleading. It would be unfair to pin down to them every recent exponent of relativity; shading of attitude there cannot fail to be. Yet they still muster heavy enough backing to call for definite abatement. The change of "rest-mass" with internal energy is set off as a distinct question.

§ Equation (10), below.

indeterminateness now; while the fullest employment of their united suggestions must mark the sanest course in adjudicating questions there, or in regard to gravitation. In that interest, this paper turns solely to detecting simple links of interrelation between the two methods. Whatever yokes standard dynamics and relativity for joint service based on complete reciprocal consistency-this the new line of approach seeks. Preferential choice may nevertheless remain open to individual opinion; or indeed it may prove to shift with the class of problem in hand. Some resettlement of rating in this quarter is scarcely avoidable, because the extension to variable inertia certainly upsets at least one allegation. The fact looms up at once that the rigid dynamics (of Einstein's theory) becomes now a convergence-point of two more general forms, under approximations through_dm/dt=0; c=∞x; respectively. It is not tenable that broader Newtonian dynamics itself reduces directly from relativity by neglecting in the latter terms that contain powers of (1/c)

*

Those expressions for tangential force, activity, and work laid down in the previous paper lie then at the heart of the matter. For convenience we quote them, with minor changes :

d

dt

d

T2 = (Qo) =

dt

d

dm

(2)

dt

2

dm

d t

dt.

The subscript letter (o) tags a quantity, here and elsewhere, as observed," or as calculated without artifice from observations. The equations are written for one standard frame (F), always for standard signs and in C.G.S. measure. Plainly (Q.) symbolizes momentum, and (E) kinetic energy. So long as the dynamical process mirrors quantitatively one of pure mechanics, nothing hinders a mixed reading for (m); it can cover narrow weight-mass and inclusive inertia, with (Qo, E。) read to correspond.

In equations (2, 3), the "Principle of vis viva" has gone by the board, along with an exclusive measurement of force

* Cf. Silberstein, p. 115; Laue, Das Relativitätsprinzip, p. 157; and

others.

Phil. Mag. Ser. 6. Vol. 40. No. 235. July 1920. D