k étant un nombre pair >1. Si k est impair et 1, la quantité entre les x crochets s'évanouit, parceque le développement de ne contient pas ex. 1 des puissances impaires de x d'un exposant > 1. relation S. 4. Développement de [(1+x)]". La formule du binôme peut être transformée aisément à l'aide de la (1+x)" = 1 +¿(1 + x) * + [2 (1 + x) } 1.1/2 + · · · · les coefficients des mêmes puissances de u peuvent être égalisés. Cela donne Cette formule peut servir à transformer une série de la forme al(1+x)+b[l(1+x)]2+c[l(1+x)]2 + · · ... en une autre qui ne contient que des puissances de x. On a par exemple Les deux conditions énoncées pour l'équation (2.) peuvent être réduites à une Passons à une autre application de la formule (1.). De l'intégrale Maintenant la fonction [—7(1-x)]"-1 peut être développée. En intégrant les termes de la série, on obtient 1 n n+1 a(a+1) ... (a+n−1)+a(a+1)... (a+ n) + a(a+1) ... (a + n + 1) + · it .... Ce résultat, dont ont fait usage dans le calcul inverse des différences, a été déjà donné par Stirling. S. 5. Développement de D" f(lx). La différentiation réîterée de la fonction f(la) donne les expressions Df(lx) = — f'(lx), D2f(lx) = }}[f" (lx)—ƒ' (lx)], D2f{lx) = }}[f(lx)—3f" (lx)+2f'(lx)], On en conclut que l'expression générale est de la forme (1.) En même temps on voit, que les coefficients Â, Â, Â1⁄2,... qui sont à 09 n déterminer, ne dependent pas de la nature de la fonction f. Leurs valeurs peuvent dont être trouvées si l'on prend une fonction f telle, que les différentiations à droite et à gauche dans l'équation ci-dessus sont praticables. La plus simple fonction de cette espèce est f(y)=e-By, et on en obtient f(k) (y) = (−1) ẞke ̄3y, f(1) (lx) = (−1)* ß*x-*, parcequ'on a immédiatement (B+ n − 1) x ̄ß—n ̧ f(lx) = x2 et D'f(lx)=(-1)^ ß(B+1)(B+2) . . . (ß+ n − La formule (1.) donne maintenant en substituant les valeurs mentionnées: ... 77 Cette équation fait voir que les coefficients A, A1, A2, sont ceux de la faculté (+1)"; donc on obtient la formule remarquable suivante: (2.) D'f(lx) = — [Ĉƒ‹(lx) — Ĉ‚ƒ‹1−1)(lx)+¶‚ƒ¤−2(lx) — · · ·]. Pour faciliter les calculs numériques, nous ajoutons une petite table des coefficients C et C. 22. An Essay on the Application of mathematical Analysis to the theories of Electricity and Magnetism. (By the late George Green, fellow of Gonville and Cains-Colleges at Cambridge.)*) General preliminary results. 1. The function which represents the sum of all the electric particles acting on a given point divided by their respective distances from this point, has the property of giving, in a very simple form, the forces by which it is solicited, arising from the whole electrified mass. We shall, in what follows, endeavour to discover some relations between this function, and the density of the electricity in the mass or masses producing it, and apply the relations thus obtained, to the theory of electricity. Firstly, let us consider a body of any form whatever, through which the electricity is distributed according to any given law, and fixed there, and let x', y', z', be the rectangular co-ordinates of a particle of this body, o' the density of the electricity in this particle, so that dr'dy'dz' being the volume of the particle, q'dx'dy'dz' shall be the quantity of electricity it contains: moreover, let' be the distance between this particle and a point p exterior to the body, and V represent the sum of all the particles of electricity divided by their respective distances from this point, whose co-ordinates are supposed to be x, y, z, then shall we have the integral comprehending every particle in the electrified mass under consideration. *) Vide tome 39. p. 13 of this Journal. Laplace has shown, in his Mec. Celeste, that the function V has the property of satisfying the equation and as this equation will be incessantly recurring in what follows, we shall write it in the abridged form 0=SV; the symbol & being used in no other sense throughout the whole of this Essay. In order to prove that 0=SV, we have only to remark, that by differentiation we immediately obtain 0=8, and consequently each element of V substituted for V in the above equation satisfies it; hence the whole integral (being considered as the sum of all these elements) will also satisfy it. This reasoning ceases to hold good when the point p is within the body, for then, the co-efficients of some of the elements which enter into V becoming infinite, it does not therefore necessarily follow that V satisfies the equation 0 = SV, although each of its elements, considered separately, may do so. In order to determine what SV becomes for any point within the body, conceive an exceedingly small sphere whose radius is a inclosing the point p at the distance b from its centre, a and b being exceedingly small quantities. Then, the value of V may be considered as composed of two parts, one due to the sphere itself, the other due to the whole mass exterior to it: but the last part evidently becomes equal to zero when substituted for V in SV, we have therefore only to determine the value of SV for the small sphere itself, which value is known to be δ(2παρ - πιρ); being equal to the density within the sphere and consequently to the value of ' at p. If now x, y, z,, be the co-ordinates of the centre of the sphere, we have |