for all the results obtained, while neglecting it entirely would cause a still greater divergence between theory and observation. On account of the thickness of the bundle of ten plates of glass, a portion of the secondary reflection would be thrown a considerable distance to one side, especially when i is large, so that it might fall quite outside of the instrument, or even be cut off by the ends of the plates. This effect would be least marked with the polarimeter, next with the Savart, and most of all with the optical circle, on account of the small aperture of the telescope. But this is just the order in which the observations stand, all of them falling between the two theoretical curves. These observations also show the effect to be expected from a bundle of plates when used to polarize light by refraction. If ten plates are employed, set, as is usual, at 57°, the polarization would be only 67.2 per cent. if internal reflection took place, but would be 95.2 if this were in any way excluded. We may in passing point out that an advantage might be expected in such a polariscope from an increase in the angle of incidence, the increased polarization probably more than making up for the loss of light and distortion induced by the increased obliquity of the incident rays. The want of perfect transparency of the glass would also tend to increase the polarization by enfeebling the secondary reflection; and dirt or grease on the surface of the glass would produce the same effect. With eight or with two surfaces these disturbing causes are much less marked, except for large angles of incidence, and hence the agreement with theory much better. XVII. Remarks on the Analytical Principles of Hydrodynamics, in Reply to Professor Challis. By ROBERT MOON, M.A., Honorary Fellow of Queen's College, Cambridge*. WE WHILE appreciating the candour evinced by Professor Challis throughout our discussion, I cannot suffer to pass in silence his assertion that my "hydrodynamical researches are founded on differential equations which are really the same as those employed by Mr. Earnshaw." Professor Challis's argument to prove this is founded upon a complete fallacy. For, taking part of my solution of the equa tion * Communicated by the Author. See Phil. Mag. vol. xlvii. p. 27. (1) he says:"Putting on the left-hand side of this equation Ap, v) for p, it will be seen, since x(v+2) is arbitrary in forin and value, that v is an arbitrary function of p, and, by consequence, on substituting for v in fp, v), that p is also an arbitrary function of p." The step here taken is nothing other than eliminating p between (2) and the equation p=f(p, v), where f(p, v) is absolutely identical with the right-hand side of (2); and no such result as funct. (p, v)=0, or v= funct. P, can be derived from such elimination. For, when x in equation (2) is spoken of as representing an arbitrary function, all that is meant is that in every case of motion the pressure may be expressed in terms of p and v in the manner indicated by (2), where x represents some function or other depending upon the particular circumstances of the motion. In any particular case of motion x will have a definite value; and to assume that in this particular case another relation exists between p, p, v, viz. p=f(p, v), where f(p, v) is not identical with the right side of (2), is simply to beg the question at issue. To put the matter in a different light-we have to start with, as Professor Challis admits, p=f1(x,t), p=ƒ2(x,t),_v=f3(x, t). How can it be possible, in general, to eliminate three quantities between these three equations so as to obtain the result v=funct. p, or p= funct. p? It is true that under particular circumstances we may do this, and under those circumstances no doubt Mr. Earnshaw's equations and mine may coincide; but the ratio of the number of cases in which our equations differ to the number in which they agree must be simply infinite. I observe that Professor Challis refers to my solution of (1) as a solution. I contend that it is the solution-the only one which the equation admits of. I have no doubt that this could be shown directly, in the manner I have elsewhere adopted to show that Poisson's integral of the accurate equation of sound for motion in one direction is the most general solution of that equation*; but it may be proved much more compendiously as follows. The solution consists of the three equations *See Phil. Mag. for August last. + S (@) '(w)-2a a α D in which the three functions p, t, are arbitrary. But in any particular case of motion, p, p, and v must follow some given law, i. e. must be expressible at a given time by definite functions of x; from which circumstance, as I have elsewhere shown, the forms of p, 1, 2 can be determined; and, being so determined, the last three equations will represent the motion throughout the entire period of its continuance. And there can be no relations between p, p, v, x, t in the case of motion under consideration other than those above given; for if there were such, it would follow that definite values of the pressure, density, and velocity at a given time admit of those quantities having alternative, and therefore ambiguous, values during the subsequent motion; which is impossible. I 6 New Square, Lincoln's Inn, January 17, 1874. XVIII. On Galvanic Polarization in Liquids free from Gas. By Dr. HELMHOLTZ*. TAKE leave to communicate to the Academy the results of a series of experiments which I have made upon the gal vanic polarization of platinum. The experiments mostly required a very long time; and I therefore crave pardon if I am obliged to leave unanswered for the present a number of further questions which at the same time obtrude themselves. It is known that when a Daniell's zinc-copper element is closed by a water-decomposition cell with platinum electrodes, a current arises of quickly diminishing force, which, in the usual way of arranging the experiment, becomes indeed after a short time very feeble, but does not entirely cease even after a very long time. We will call this the polarizing current. afterwards separate the decomposition cell from the Daniell's element, and connect its platinum plates with the voltameter, we obtain another (the depolarizing) current, which in the cell is opposite in direction to the polarizing current, and is likewise If we * Translated from the Monatsbericht der königlich preussischen Akademie der Wissenschaften, 1873, pp. 587–597. Phil. Mag. S. 4. Vol. 47. No. 310. Feb. 1874. powerful at the commencement, under the ordinary conditions of observation, but mostly soon dwindles to imperceptibility. It is this simple experiment to which my investigations refer. The question to be solved was, On what depends the apparently unlimited duration of the polarizing current? for in a series composed as above described, unless other changes also occur in it, the electrolytic conduction which results according to Faraday's law in the liquids cannot take place without infringing the law of the conservation of energy. That is to say, if no other equivalents of potential energy are expended, in such a series the mechanical equivalent of the heat generated in the circuit must be equal to the work equivalent of the chemical forces which become active and are expended in electrolysis. The latter, however, if the decomposition proceeds according to the law of electrolytic equivalents, is negative*, and therefore cannot be equal to a positive heat-work to be produced by the current. If, then, Faraday's law is exclusively valid, the decomposition of water, even in the minutest quantity, cannot be maintained continually by a Daniell's element. Indeed no disengagement of the gases of which water is composed is observed in the above-described experiment, however long the current lasts. It is moreover to be remarked that not even by diffusion or any process similar to diffusion could the molecules of hydrogen and oxygen, which on the polarization of the plates are urged towards them, become free and, again unelectric, remove from them. Such an occurrence would finally give, as the result of the work, a water-decomposition for which no equivalent inciting force would be present in the Daniell's element. If, as is probable, on the polarization of the electrodes a changed arrangement of the atoms of hydrogen and oxygen enters, whether in the interior or at the boundary surfaces of the liquid, these particles are at all events kept in their places by chemical or electric forces of attraction till new forces come to aid of sufficient power to set them free. Whatever relations between electric and chemical attraction-forces may be assumed, if the law of the conservation of energy holds good, an electric attraction upon one of the elements, whose potential is sufficiently great to overcome the chemical affinity, can itself be again overpowered * According to Andrews, 1 gramme of hydrogen in burning to water gives 33,808 heat units; according to Favre and Silbermann, 34,462. For each gramme of hydrogen, in the Daniell's element, 325 grm. of zinc are dissolved, and the equivalent amount of copper precipitated. This quantity of zinc, when it separates copper from combination with sulphuric acid, generates, according to Favre, only 23,205 units of heat. Corresponding to this, an electromotive force of at least 1 Daniell's element is requisite in order to maintain even the feeblest lasting decomposition of water. |