June 12, 1856. The LORD WROTTESLEY, President, in the Chair. The following gentlemen were admitted into the Society :- Philip Henry Gosse, Esq. Archibald Smith, Esq. The following communications were read: "On the Construction of the Imperial Standard Pound, and its copies of Platinum; and on the comparison of the Imperial Standard Pound with the Kilogramme des Archives." By W. H. MILLER, M.A., F.R.S., Professor of Mineralogy in the University of Cambridge.—Part II. Received June 7, 1856. (Abstract.) The Quartz Weight The hardness of quartz, its capability of taking a high polish, the absence of any hygroscopic properties, and its indestructibility at the ordinary temperature of the atmosphere by any chemical agent except hydrofluoric acid, are such valuable qualities in a substance used for the construction of weights, that Professor Steinheil adopted it as the material for a copy of the kilogramme. The only objection to the use of a weight made of quartz is, that on account of the large amount of air displaced, the barometer and thermometer must be observed with extreme care during its comparison with a weight made of any ordinary metal. The Committee commissioned Mr. Barrow to construct a weight of quartz sufficiently near to 7000 grs. to admit of readily deducing the pound from it. Its form is that of a cube of about 2.2 inches, having its edges and angles rounded. properties if the stress at any point of it was influenced only by parts of the body touching it; but if the stress at one point is directly influenced by the strain in parts at distances from it finite in comparison with the wave length, the helicoidal property might exist, and the rotation of the plane of polarization, such as is observed in many liquids and in quartz crystals, could be explained as a direct dynamical consequence of the statical elastic reaction called into play by such a strain as exists in a wave of polarized light. It may, however, be considered more probable that the matter of transparent bodies is really heterogeneous from one part to another of lineal dimensions not infinitely small in comparison with a wave length, than that it is infinitely homogeneous and has the property of exerting finite direct "molecular" force at distances comparable with the wave length and it is certain that any spiral heterogeneousness of a vibrating medium must, if either right-handed or left-handed spirals predominate, cause a finite rotation of the plane of polarization of all waves of which lengths are not infinitely great multiples of the steps of the structural spirals. Thus a liquid filled homogeneously with spiral fibres, or a solid with spiral passages through it of steps not less than the forty-millionth of an inch, or a crystal with a righthanded or a left-handed geometrical arrangement of parts of some such lineal dimensions as the forty-millionth of an inch, might be certainly expected to cause either a right-handed or a left-handed rotation of ordinary light (the wave length being th of an inch for homogeneous yellow). But the magnetic influence on light discovered by Faraday depends on the direction of motion of moving particles. For instance, in a medium possessing it, particles in a straight line parallel to the lines of magnetic force, displaced to a helix round this line as axis, and then projected tangentially with such velocities as to describe circles, will have different velocities according as their motions are round in one direction (the same as the nominal direction of the galvanic current in the magnetizing coil), or in the contrary direction. But the elastic reaction of the medium must be the same for the same displacements, whatever be the velocities and directions of the particles; that is to say, the forces which are balanced by centrifugal force of the circular motions are equal, while the luminiferous motions are unequal. The absolute circular motions being therefore either equal or such as to transmit equal centrifugal forces to the particles initially considered, it follows that the luminiferous motions are only components of the whole motion; and that a less luminiferous component in one direction, compounded with a motion existing in the medium when transmitting no light, gives an equal resultant to that of a greater luminiferous motion in the contrary direction compounded with the same non-luminous motion. I think it is not only impossible to conceive any other than this dynamical explanation of the fact that circularly polarized light transmitted through magnetized glass parallel to the lines of magnetizing force, with the same quality, right-handed always, or left-handed always, is propagated at different rates according as its course is in the direction or is contrary to the direction in which a north magnetic pole is drawn; but I believe it can be demonstrated that no other explanation of that fact is possible. Hence it appears that Faraday's optical discovery affords a demonstration of the reality of Ampère's explanation of the ultimate nature of magnetism; and gives a definition of magnetization in the dynamical theory of heat. The introduction of the principle of moments of momenta ("the conservation of areas") into the mechanical treatment of Mr. Rankine's hypothesis of " molecular vortices," appears to indicate a line perpendicular to the plane of resultant rotatory momentum ("the invariable plane") of the thermal motions as the magnetic axis of a magnetized body, and suggests the resultant moment of momenta of these motions as the definite measure of the "magnetic moment." The explanation of all phenomena of electro-magnetic attraction or repulsion, and of electro-magnetic induction, is to be looked for simply in the inertia and pressure of the matter of which the motions constitute heat. Whether this matter is or is not electricity, whether it is a continuous fluid interpermeating the spaces between molecular nuclei, or is itself molecularly grouped; or whether all matter is continuous, and molecular heterogeneousness consists in finite vortical or other relative motions of contiguous parts of a body; it is impossible to decide, and perhaps in vain to speculate, in the present state of science. I append the solution of a dynamical problem for the sake of the illustrations it suggests for the two kinds of effect on the plane of polarization referred to above. Let the two ends of a cord of any length be attached to two the problem of determining the function which (f, F being given functions, and the limits a, ẞ of the integration being also given) satisfies the equation S® f(x, 0) 9(x+0) d0 = Fx. He observes, that, unlike the methods employed in his former memoir, and the solutions there employed, which are quite rigorous, the methods of the present memoir depend upon developments into series, the strictness of which has been contested by some mathematicians; but that passing over these difficulties, he has solved the famous problem, the solution of which has been vainly sought after for the last two hundred years, because on the above-mentioned equation depends the integration of the generally linear equation of any order whatever of two variables, and consequently the whole Integral Calculus. The solution first obtained by the author, and which he afterwards exhibits under a variety of different forms, is as follows: Theorem I.-The equation being given, "f(x, 0) 4(x+0) d0 = F,(x) = F(x), where f(x, 0) is a given function of x and 0; F(x) is a given function of a such that the equation F(x)=x cannot hold good for any finite value of a; F(x) a given function of a containing all the factors which render F(x) infinite, and the function F(x) being absolutely arbitrary; and a and ß being given constants (independent therefore of a and 6), the expression for a which satisfies the preceding equation is where f(x) is determined by f(x) = (x−a,) e††(a,) (a,) is a root of the equation α B 1 S e1‚o f(a,, 0) do :0 solved relatively to m, and a,, a, a,, &c. are the roots of F1(x)=0. The author afterwards considers the equation S'f(x, 0) 9 (0) d0 = F(x), α and the solution of a linear equation is at once made to depend upon this as follows: viz. given for the determination of the function x the equation do (x) f(x, 0) 9(x) +ƒ (x, 1) + &c. = F(x). dx a, ß being constants, and (0) a function of 0 to be determined. It is always permitted to assume this equation. By this means, writing for shortness f(x, 0) = f(x, 0) + f(x, 1) 0 + &c., A solution which occurred to the author after the memoir was drawn up, is as follows: viz. given, as before, the equation (^ƒ(x, 0) 9(x+0) d0 = F(x), then (w√1 being determined by the equation |