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TABLE VI.--Heat of Combination of the Compounds in the Liquid and Solid Condition.
* Da represents the heat of dissolution of the water, naphthalene, or benzene, Do heat of the other constituents, and Dab that of the compound.
LI. On Helmholtz's Electrochemical Theory, and some
To the Editors of the Philosophical Magazine.
ROFESSOR G. JOHNSTONE STONEY, in his paper on the "Electron," or Atom of Electricity, in the Philosophical Magazine of October 1894, very rightly draws attention to the fact that he expressed himself first, with regard to Faraday's law, at the Belfast Meeting of the British Association in August 1874, as follows:-"For each chemical bond which is ruptured within an electrolyte a certain quantity of electricity traverses the electrolyte, which is the same in all cases."
Professor G. J. Stoney calls this smallest quantity of electricity the "Electron," and estimates it at 3 x 10-11 of the C.G.Š. electrostatic unit of electricity.
In this view, therefore, he anticipated Helmholtz in his Faraday Lecture in April 1881. Helmholtz, however, then propounded further the hypothesis that, "in the case also of non-electrolytes, the Valencies' are charged with the same atoms of electricity." Helmholtz explains, moreover, the grounds for the supposition that the attraction between the electrons is the most essential and the greatest part of chemical force. The old electrochemical theory of Berzelius acquired herewith an entirely new form through Helmholtz in respect of the quantity of the atom charges, and deserves therefore the title of "Helmholtz's Electro-chemical Theory.'
Without knowing Prof. G. Johnstone Stoney's calculation of the "Electron," I also, in a paper "Ueber die electrischen Kräfte der Atome," read before the Niederrheinische Gesellschaft für Naturkunde on the 1st Dec. 1890 and 12th Jan. 1891*, calculated the electron, and first attached thereto calculations fitted to decide whether "the forces operating in the atoms of a molecule have the same order of magnitude as the electrostatic attraction of the valency-charges.'
That this is the case I had then already proved, in respect to the dissociation heat N2O4 into 2NO2, and I, into 21.
I have further assumed that both atoms of a molecule revolve round each other with a constant velocity, which is given by Boltzmann's kinetic theory of polyatomic gases.
*F. Richarz, Sitzungsberichte, Bonn, vol. xlvii. p. 113 (1890); vol. xlviii. p. 18 (1891).
The equation that the centrifugal force is equal to the force of attraction gives for the latter a value nearly equal to that of the attraction of two "electrons" on each other. Further, I put forward at the same time the hypothesis that radiation is caused by the oscillations of the valency-charges. This supposition was not new, as I have since found. His own quotation in his book, 'Theorie der electrischen und optischen Erscheinungen in bewegten Körpern,' Leiden, 1895, page 5, called my attention to the fact that Prof. H. A. Lorentz as early as 1878 attributed light-waves to electrical particles, which are joined to the atoms and which are also assumed in electrolysis [Verh. d. kgl. Akad. v. Wetenschappen, 18 Deel, Amsterdam, 1879; notice especially the conclusion, page 112]. Professor Hertz, as he told me, was of a similar opinion; however, he was not attached to the electrochemical theory of Helmholtz, but to the opinion of Victor Meyer and Riecke (Berliner Chem. Ber. xxi. p. 946, 1888). But Lorentz's electromagnetic theory of refraction is, like Helmholtz's theory of Dispersion (1892), independent of the size of the valency-charges. From this quantity, the "electron,” I have, in my paper mentioned above of the 12th Jan. 1891, calculated that the period of rotation of the two atoms of a molecule round one another is about 10-14 seconds. This is the period of the electrodynamic radiation which the electrons of the two atoms give out while they rotate together with the ponderable atoms round one another. It would correspond with ultra-red waves. As the computed value of the period of rotation is only the average value of the different periods of rotation possessed at the same time by different molecules, the emission must give a more or less extended spectrum of dark heat-rays, which spectrum would have its maximum in the region of the average value. Indeed emission of gases is similar, in so far as the latter is only based on increase of temperature. When the period of rotation is accelerated the spectrum might possibly pass into the visible region.
Prof. G. J. Stoney took up the matter of the electrodynamic radiation of oscillating electrons at about the same time as I did, but in other respects (Trans. Roy. Dublin Soc. vol. iv. 1891, p. 585); later also Prof. H. Ebert (Arch. de Genève,  xxv. p. 489, May 1891).
I next developed the purely kinetic part of my conclusions (Wied. Ann. xlviii. March 1893, pp. 467-492). There I have also taken into consideration the dissociation-heat of hydrogen given by Prof. Eilhard Wiedemann. The same I have also made use of for comparison, besides the dissociation
heat of nitric tetroxide and iodine, already made use of in the first publication of 12th Jan. 1891, in the detailed accounts of my electrical calculations (Münchener Akademie, Bd. xxiv. p. 1, 13 Jan. 1894, and Wied. Ann. Bd. lii. p. 385, 1894).
There I have also added the following calculation. Assuming that molecular magnetism is produced by the rotation of the valency-charges, we obtain for the specific magnetism at saturation-point values which correspond in the order of magnitude with those found by experiment.
Helmholtz's electrochemical theory has meanwhile also been confirmed in other respects by the very interesting calculations of Prof. A. P. Chattock (Phil. Mag.  xxxii. p. 285, 1891; xxxiv. p. 461, 1892; xxxv. p. 76, 1893). I am, Gentlemen,
University of Bonn, April 1895.
LII. Note on a Simple Graphic Illustration of the Determinantal Relation of Dynamics. By G. H. BRYAN*.
N the whole range of theoretical dynamics there is probably
determinantal relation connecting the multiple differential of the initial coordinates and momenta of a system with that of its final coordinates and momenta. This relation, which may almost be regarded as the keystone to the Kinetic Theory of Gases, is conveniently written in the Jacobian form
where P1, P2,
are the generalized momenta corresponding to the generalized coordinates 1, 2,..., and unaccented and accented letters refer respectively to initial values and final values after a fixed interval of time t.
To my mind the difficulty of grasping this result arises from the want of simple graphical illustrations and verifications from first principles not involving the use of the Calculus. The following illustrative examples of its applications to systems with one degree of freedom have afforded me great assistance in understanding the theorem, and I trust that they may prove useful to others.
Consider a particle moving in the straight line OX (fig. 1)
*Communicated by the Physical Society: read April 26, 1895.
under any law of force. Let M be the position of the particle at any instant, and let the velocity of the particle at this instant be represented by the ordinate MP drawn at right angles to OX. Then, since the momentum is proportional to the velocity, the coordinates OM, MP represent the coordinate and momentum of the particle at the given instant, and we may call P the representative point.
Now let four such particles of equal mass be projected simultaneously, having the initial coordinates x and x+dx and the initial velocities v and v+8v. The representative points will form a small rectangle PQRS of area da. Sv.
Let P'Q'R'S' be the corresponding representative points at any subsequent instant t.
Then the determinantal relation asserts that the area of the small parallelogram P'Q'R'S' is equal to that of the rectangle PQRS.
[Instead of taking four particles we might suppose the points P, Q, R, S to refer to the same particle projected with different initial conditions and allowed to move for a fixed time-interval t.]
This property may be verified from first principles in the following simple cases :
CASE I. Let the motion be uniformly accelerated. Then from the equations
v'=v+ft, x'=x+vt + 1 ft2,
and the corresponding equations obtained by substituting x+dx for x and v+dv for v, it is easy to see (fig. 1) that the
parallelogram P'Q'R'S' has its base P'Q' parallel to OX and equal to PQ or Sa, and its altitude equal to PQ or Sv. Therefore
area P'Q'R'S' area PQRS.
The parallelogram will, however, have undergone a shear,