It cannot be surprising that the values of the reflection-constants change with the distance of the opaque particles, since the transparency and colour of thin layers of the same metal also depend on it. (Faraday, Exp. Res. iv. 391.) The dependence of the optical properties of a quantity of more or less transparent and non-transparent particles on the distance of the latter seems to me hardly compatible with the assumption that absorption and dispersion of light are determined solely by a mutual vibration of the molecules of the body, occasioned by the motion of the æther. Würzburg, Aug. 20, 1873. XL. On Sylvester's and other forms of continued Fraction for Circle-quadrature. By THOMAS MUIR, M.A., F.R.S.E., Assistant Professor of Mathematics in Glasgow University*. N the Philosophical Magazine for May 1869, Professor Sylvester gave as a deduction from a solution of an equation in finite differences the following notable identity : The origin of it was so peculiar, that to the reader it could not but seem a desirable thing to establish a more general result, and point out some connexion between it and previous special results of the same kind. This has been done in one direction by Mr. J. W. L. Glaisher, in a paper recently read before the Mathematical Society of London, in which he transforms the product (1 + 2) (1 + 2) (1 + 2)... into a series, this series into a continued fraction, and then points out that, from the identity of the product and the continued fraction, Professor Sylvester's result directly follows as a particular case. The object of the present short paper is to establish another such line of relationship. * Communicated by the Author. (n + 1) $ (n + 1 ) Now the denominator on the right is the same function of n+1 that the left-hand member is of n; and thus by continued substitution we have If n be now taken equal to 1, the left-hand member here becomes Putting in this general result m=1, we obtain, as desired, and so on. But, secondly, by expanding ym-1(1+ y)-1 in a series of ascending powers of y, and integrating with respect to y between the limits 0 and 1, we obtain Now the series here enclosed in brackets may be looked on as a particular case (viz. a=m, B=1, y=m+1, x=-1) of Gauss's general series F(x, ß, y, x), which, owing to a special property, he was able to expand in the form of a continued fraction. Taking advantage of this, we find, on making the necessary substitutions, and, as before, putting m=, we derive the well-known result, π 2 12 22 3+ 32 5+ 7+. (B) and putting m=1, there results a companion identity to (a), and so on. Thirdly, the series found above may be expanded in the form of a continued fraction by means of the ordinary general method. Doing this, it is found that whence, by the substitution of for m, we obtain 2 π 2 12 32 5o 2+ (2) which takes us back to Brounker and the infancy of continued fractions. The University, Glasgow, March 28, 1874. P.S. Since the above was written I have read part of an encyclopædic paper by Stern in Crelle's Journal, vols. x. and xi. (1832-33), entitled "Theorie der Kettenbrüche und ihre Anwendung," and have to-day arrived with surprise at the third chapter, the first division of which has the same title as Mr. Glaisher's paper above referred to, viz. "Verwandlung der unendlichen Producte in Kettenbrüche." In this Stern establishes a general formula of transformation, and then proceeds to give examples illustrative of it. Among these are five forms of continued fraction for and first of the five the very fraction we have been speaking of as Sylvester's. Stranger still, in a footnote Stern adds, "Diesen Ausdruck hat schon Euler auf anderem Wege gefunden" (Com. Ac. Petr. vol. xi. p. 48); so that the expression derived so ingeniously in 1869 must at that time have been over a century old. Professor Sylvester and Mr. Glaisher will, I am sure, rejoice with other workers to see justice done to their predecessors Euler and Stern. π 2' The University, Glasgow, April 10, 1874. XLI. On the Physical Constants of Hydrogenium. N March 1869 I communicated to the Society a paper entitled "Motion of a Palladium Plate during the Formation of Graham's Hydrogenium," which appears in the Proceedings for Session 1868-69 †. When engaged with this subject, many points of interest regarding the behaviour of palladium containing occluded hydrogen suggested themselves for investigation; and in concluding the paper I remarked that "careful determinations must be made of the electromotive force, latent heat, &c. * From the Proceedings of the Royal Society of Edinburgh, Jan. 20, 1873. Communicated by the Author. † Phil. Mag. June 1869, pp. 424-431. of hydrogenium" before we could arrive at any conclusion regarding the condition of the absorbed hydrogen. Subsequently Professor Tait made a series of determinations on the "Electrolytic Polarization of Palladium Electrodes"*, devising a new and ingenious method for the purpose. Although at different times subsequent to my first communication the problem of determining the physical constants of hydrogenium recurred, as my attention was in the meantime directed to the specific heat of carbon at high temperatures, no progress was made with the investigation until September 1872, when the results of my preliminary experiments were communicated to the Philosophical Magazine, under the title "Note on the Specific Heat of Hydrogenium.' In that note it is stated that by means of a specially constructed calorimeter the specific heat of hydrogen in palladium is found to be 3.1 per atomic weight, nearly identical with that of gaseous hydrogen. The present paper deals with some of the physical constants of hydrogenium, more especially with the specific gravity, specific heat, and coefficient of expansion. Graham, in his celebrated paper on hydrogenium, made many determinations of the specific gravity of the occluded hydrogen by observing the increase of length of palladium wire after being fully charged, thus finding the cubical expansion, and from it deducing the weight of unit volume of the absorbed hydrogen. From experiments made in this way he found the specific gravity to be nearly 2. Afterwards he discovered the value was about three times what it ought to be, from a contraction of length occurring when palladium wire is used. This he confirmed by the use of alloys of palladium that resist this contraction, and finally regarded the specific gravity as 0.733. No determinations were made when the palladium was partially saturated; and he rejected the ordinary process of taking specific gravities, because of the continual evolution of gas preventing exact weighings being taken in water. In the experiments to be detailed a cubical mass of the metal was charged with hydrogen by electrolysis, taken at different times during the progress of the saturation and weighed in air and in water. If the mass was allowed free exposure to the air for several hours, little difficulty arose from the evolution of gas when immersed in water, and accurate results could be obtained. From the experimental numbers the specific gravity of the absorbed hydrogen was calculated by the well-known formula * Proceedings of the Royal Society of Edinburgh, Session 1868-69. |