conditions to be satisfied in integrating the equations of motion would be simply that the initial distribution of electricity along each must be whatever is prescribed; that is, that and denoting two arbitrary functions. Hence, according to Fourier, we have, for the integrals of the equations (3), and the solution of the problem is expressed in terms of these integrals by (4). If now we suppose the cable to have one end at a finite distance from the part considered, for instance at the point O from which a is reckoned, and if at this end each wire is subjected to electric action so as to make its potential vary arbitrarily with the time, there will be the additional condition is to be fulfilled. In the other conditions, (5), only positive values of a have now to be considered, but they must be fulfilled in such a way as not to interfere with the prescribed values of the potentials at the ends of the wires; which may be done according to the principle of images, by still supposing the wires to extend indefinitely in both directions, and in the beginning to be symmetrically electrified with contrary electricities on the two sides of O. To express the new condition (7), a form of integral, investigated in a communication to the Royal Society (Proceedings,' May 10, 1855, p. 385), may be used; and we thus have for the integrals of equations (3), Lastly, instead of the cable extending indefinitely on one side of the end O, let it be actually limited at a point E. If the ends of the if x and x denote two arbitrary functions, and a the length OE. Either of these requirements may be fulfilled in an obvious way by the method of successive images, and we so obtain the following respective solutions : where F, J, E, & denote for brevity the following functions: kc(x+2ia)2 €ŋŋ(z, t−0) = 2 (−1) '(x+2ia)g ̄*(1+1)(t−0) = (1+f)(t−0) kc Each of the functions F and E is clearly the difference between two periodical functions of (-) and (+); and each of the functions and E is a periodical function of a simply. The expressions for these four functions, obtained by the ordinary formula for the expression of periodical functions in trigonometrical series, are: Either (11) or (12) may be used to obtain explicit expressions for the solutions (10) and (10)', in convergent series; but of the series so obtained, (11) converge very rapidly and (12) very slowly when t is small; and, on the contrary, (11) very slowly and (12) very rapidly when t is large. It is satisfactory, that, as increases, the first set of series (11) do not cease to be, before the second set (12) become, convergent enough to be extremely convenient for practical computation. The solutions obtained by using (12), in (10) and (10)1, are the same as would have been found by applying Fourier's ordinary process to derive from the elementary integral -mt sin ne the effects of the initial arbitrary electrification of the wires, and employing a method given by Professor Stokes* to express the effects of the variations arbitrarily applied at the free ends of the wires. CASE II.-Three-wire Cable. The equations of mutual influence between the wires may be clearly put under the forms cv1 =q1+f(92+93), cvq=92+f(q3+91), cv3=93+f(91+92) ; *See Cambridge Phil. Trans. vol. viii. p. 533, “On the Critical Values of the sums of Periodic Series." (12). and the equations of electrical motion along them are then as follows: and require that w1+w2+w1=0, we find by addition and subtraction, among the equations of conduction, where for w may be substituted either w1, w, or w3. CASE III.-Four-wire Cable. The equations of mutual influence being cv1 =9x+f(92+94) +IL39 and other four symmetrical with this; and the equations of motion, = +1 dt dx2 4 1 &s=1 (0+d−2w,); &s= | (0—9—2w2); and we find from the equations of conduction, CASE IV.-Cable of six wires symmetrically arranged. Equations of mutual influence, cv1 =q1+f(92+96)+9(93+Qs)+hq+ da These equations, integrated by the usual process to fulfil the prescribed conditions, determine a, d, w1, wy, wз, P1, P2, P3; and we then have, for the solution of the problem, &i=¦ ¦ (0+&+w1+P1); 93= 12 / o+&+w2+P2); &s==+ (0+&+ws 6 "Experimental Researches on the Functions of the Mucous Membrane of the Gall-bladder, principally with reference to the Conversion of Hepatic into Cystic Bile." By George Kemp, M.D. Cantab. The author deduces from his experiments the following generali zations :- 1st, That the mucus of the gall-bladder is not merely a secretion destined to lubricate the interior of that organ and protect it from the irritation of its other contents, but is an essential integral portion of the cystic bile. 2ndly, That the gall-bladder is not merely a receptacle and reservoir for the bile, but an organ highly endowed with organic functions; and that the proper secretion of the liver is converted into cystic bile mainly through the agency of its mucous membrane. GEOLOGICAL SOCIETY, [Continued from p. 78.] December 3, 1856.-Col. Portlock, R.E., President, in the Chair. 1. "On the Volcanic Eruption of Mauna Loa in 1855-56." By F. A. Weld, Esq. Communicated by Sir C. Lyell, V.P.G.S. In a letter dated July 12, 1856, he communicated the information he had obtained respecting the late Eruption in Hawaii, and gave a detailed account of his ascent of Kilauea and Mauna Loa, with observations on the craters and on the condition of the lava-stream which had lately been ejected from a lateral opening on the latter mountain. Mr. Weld remarked also that a slight shock of earthquake had been felt on the Island of Maui, which is also of volcanic formation. Phil. Mag. S. 4. Vol. 13. No. 84. Feb. 1856. L ; |