Winds: N, 6; NE, 4; E, 1; SE, I; S, 1; SW, 7; W, Barometer: Mean height For the month. For the lunar period, ending the 17th.... For 14 days, ending the 11th (moon south). 29.927 For 14 days, ending the 25th (moon north)... .... 30.051 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. 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 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 4r generally from the simple condition 4" xa, without assuming any relation between v and n, 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 x, 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 a2 = (4*)" or 4** x = a® x. and generally 4" x 6" x by the preceding Lemma for every value of n. If, therefore, · 4" x = a" x and b" = 1, we have x=1" (2) which by introducing an arbitrary function, according to Mr. Babbage's method, becomes for the general solution of (1) whatever be the values of v and n. When = 1, this expression gives v That excellent mathematician Mr. Herschel has given in Mr. Babbage's 11th Prob. Phil. Trans. for 1815, a different expres sion for the value of ↓r; namely, ↓ x = ̃1{(−1)" ø x}; but by what we have shown, this is the solution of "r x not of 4"x = x. Putting (3) under the form for the circular root of 1, we have in which a is the semiperiphery to radius 1, and k is any integer, And because cos n a sin λ, if we expound k by" "the double sign of (5) will n ± occasion its values to circulate and to return for all magnitudes of k into the same which take place between k = 1 and k = n or n as n happens to be even or odd, or between a = o or 1 and a = n 2, the increments of a being 2. It is also evident if n be an integer, that the number of functional roots will be n, and if u be a fraction, the number of roots will be equal to the units in the numerator of this fraction; so that if ǹ be irrational or imaginary, the number of functional roots will be infinite. This solution being performed by the coefficient may be called the coefficiential solution. whatever be the value of n. Hence if b = 1, and introducing the arbitrary function which is another general solution of (1), and being obtained by means of the exponent may be called the exponential. The e exponent in this case being put under the same form as the coefficient in the former case, admits the same observations with respect to the functional roots, &c. We may combine these two solutions, and have gonstempo ent (7) vd bas eded In this solution the coefficient 1" may be, but is not necessa rily the same root as 1" the exponent; that is, the indeterminate integers k of the coefficient and exponent are not necessarily the same. Hence the number of functional roots in this expression is n. For example, if n = 2, v = 1, and x = x, the number of roots is four, This remark, therefore, destroys the opinion derived from the analogy of algebraic equations, namely, that the functional equation x = x, n being a positive integer, has as many roots only as n contains integers. It is indeed evident from the nature of arbitrary functions, that the number of functional roots is indefinite, when the arbitrary function has its full scope; but when the arbitrary function is excluded, and not in any way anticipated, the number of functional roots is the same as the number of algebraic roots of an equation of equal dimensions. In the preceding instance the arbitrary function is in part anticipated by the double solution; and hence the reason that the number of functional roots exceeds those denoted by the index. $3. If we set out with a function of the form a + bx c+ dx the 2d, 3d, 4th, &c. functions will evidently be of the same form. And because a, b, c, d, are indefinite, any function of this form may be conceived to be the 2d, 3d, or rth function of a like form; so that we may suppose a we subr we substi it is manifestly immaterial whether in the value of stitute for r the value of 'r, or in the value of tute for x the value of r; both results will be the same. Making, therefore, these substitutions, and equating the corresponding terms of the results, we obtain A |