Magendie's New Remedies, with Appendix. 5s. 6d. Manuale Medicum, or Medical Pocket Book for Students. 5s. The Young Brewer's Monitor, comprising a scientific Summary of the Art, with a Series of Cautionary Precepts, &c. 8vo. ARTICLE XIX. NEW PATENTS C. Jefferies, Havanah Mills, near Congleton, silk thrower, and E. Drakeford, Congleton, watch-maker, both in the county of Chester, for their invented method of making apparatus for the purpose of winding silk and other fibrous materials.-July 29. W. Wheatstone, Jermyn-street, St. James's, music seller, for his invention of improving and augmenting the tones of piano-fortes, organs, and euphonons.-July 29. J. Price, Stroud, Gloucester, engineer, for certain improvements in the construction of spinning machines.-Aug. 5. G. Graydon, Bath, Captain in the Royal Engineers, for inventing a new compass for navigation and other purposes.-Aug. 5. W. Johnson, Great Tothan, Essex, for inventing a means of evaporating fluids for the purpose of conveying heat into buildings for manufacturing, horticultural, and domestic uses, and for heating liquors in distilling, brewing, and dyeing, and in making sugar and salt with reduced expenditure of fuel.-Aug. 5. J. Perkins, Fleet-street, engineer, for certain improvements in propelling vessels.Aug. 9. J. Fussell, Mells, Somerset, edge-tool-maker, for his improved method of heating woollen cloth, for the purpose of giving it a lustre in dressing. Aug. 11. H. Schroder, Hackney, broker, for his invented new filter.— Aug. 11. J. Vallance, Brighton, for his improved method of abstracting or carrying off the caloric of fluidity from any congealing water (or it may be other liquids): also an improved method of producing intense cold: also a method of applying this invention so as to make it available to purposes, with reference to which temperatures about or below the freezing point may be rendered productive of advantageous effects, whether medical, chemical, or mechanical.—Aug. 28. J. Neville, High-street, Southwark, engineer, and W. Busk, Broadstreet, for certain improvements in propelling ships' boats, or other vessels, or floating bodies.-Sept. 16. The observations in each line of the table apply to a period of twenty-four hours, beginning at 9 A. M. on the day indicated in the first column. A dash denotes that the result is included in the next following observation. ANNALS OF PHILOSOPHY. NOVEMBER, 1824. ARTICLE I. On the Use of Gold Leaf as a Test of Electromagnetism. By the Rev. J. Cumming, Professor of Chemistry in the University of Cambridge. (To the Editors of the Annals of Philosophy.) GENTLEMEN, Cambridge, Sept. 21, 1824. IN the instrument which I constructed between three and four years since for the detection of minute quantities of electromagnetism, the test employed was the action of the connecting wire on a magnetised needle; I have lately applied to this purpose the reverse principle, viz. the action of a magnet upon the connecting wire by making a slip of gold leaf a part of the circuit. The instrument is readily constructed by substituting for the two slips of gold leaf in Bennet's electrometer a single slip suspended from the wire of the upper plate, and resting upon the metallic base. Though not so delicate a test of electromagnetism as the galvanoscope above alluded to, yet with even a feeble power, I find it to be very sensible to the action of a small horse-shoe magnet; and it may, perhaps, be considered as an advantage peculiar to this instrument, that it exhibits the magnetic action of the closed circuit by a modification of the same apparatus which is used for detecting the electric action of the circuit when open. I am, Gentlemen, very truly yours, J. CUMMING. New Series, VOL. VIII. Y ARTICLE II. On the Solution of 4" x = x. By John Herapath, Esq. (To the Editors of the Annals of Philosophy.) GENTLEMEN, Cranford, Oct. 5, 1824. SINCE the publication of Mr. Babbage and Mr. Herschel's beautiful researches on periodical functions, the extension of the functional calculus is become a subject of considerable interest. Among the first and most useful parts of functions stands the solution of .... (1) 4" x = x Indeed this solution is the hinge on which all further inquiries must naturally turn. Limitations, therefore, in this part, unavoidably beget limitations in the higher operations, and thus deprive the calculus of its chief excellence, unbounded generality. All the solutions of (1) I have yet met with are confined to the evaluation of x from positive integral values of n. In the following pages I have sought the value of 4 generally from the simple condition 4" x = x, without assuming any relation between v and, or any limitation to their values. This I have effected, first by indirect methods, as it has usually been done, and then by a direct process extremely simple and general. A few observations suggested by the preceding solutions are afterwards added, respecting the number of arbitrary functions in the complete solution of (1), which it is hoped will settle that important question. Lemma.—If 4" x = a x, a x being any function of x; then 4′′ x = a2 x and 4" x = a* x, whatever be the values of p, n, v. For since is supposed equal to a x for all values of r, must be of the same form as a; and, therefore, any operation on as a whole by whatever index denoted must be identical with the same operation on a. That is a" = (4")" or "xa" x. and generally 4" x = b" x by the preceding Lemma for every value of n. If, therefore, |