the end 0, let it be actually limited at a point E. If the ends of the if X1 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, F, E, E denote for brevity the following functions : En(, t-0)= (-1)'(x + 2ia)= 4(1+f)(0–0) Each of the functions F and E is clearly the difference between two periodical functions of (&—x) and (& +x); and each of the functions F and E is a periodical function of x simply. The expressions for these four functions, obtained by the ordinary formulæ for the expression of periodical functions in trigonometrical series, are as follows: 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 f 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)', are the same as would have been found by applying Fourier's ordinary process to derive from the elementary integral e-mt sin nx 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 cv,=9. +f9z+9), cv,=9, +f(93+1), cv,=9s +(9.+92); and the equations of electrical motion along them are then as follows: d’92, d'93 d'93, dag. + tf dx2 dx2 dx2 kc d93_d=93 f dt dx2 ke d91_dog da 91 = If we assume r=91 +92 +93, w=291-9-93, w=292-93-91, ws=299-9-92) which give and require that w, +w,+w=0, we find by addition and subtraction, among the equations of conduction, d'o d.x2 and dw =(1-f) dt where for w may be substituted either W., W,, or wg. CASE III.-Four-wire Cable. The equations of mutual influence being co,=a+f(92+9.) +993, and other four symmetrical with this; and the equations of motion, d9d994 d'93 dix? dx23 +9 * See Cambridge Phil. Trans. vol. viii. p. 533, “On the Critical Values of the sums of Periodic Series.” we may assume 91 +92 +93 +9450, 9-93=wi, which give 9 = (+9+2w.); 4= (-$+2w.) and we find from the equations of conduction, d29 dt dw đưw. d.x" din?; kc da? ke dgi _ da d 94 CASE IV.-Cable of six wires symmetrically arranged. cv;=q1 + f(92 +96) +9(93+9)+hq. &c. Equations of conduction, +90*9*+d993) +20° dir? Then assuming 9 +92 +93 +94 +95 +96=0 3(91-91)-9=pi 3(93-96)-9=Pz; 3(93-92)-9=ps; which require that Wi+w, +w=0, and Pu+Pa+p==0); we have do do dag kc kc di ( dt dw ď w kc ] dt dt These equations, integrated by the usual process to fulfil the prescribed conditions, determine o, J, W., W., W3, P1, P2, Ps; and we then have, for the solution of the problem, dx2 dx2 V. “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. Communicated by the Rev. W. CLARK, M.D., F.R.S., Professor of Anatomy in the University of Cambridge. Received May 1, 1856. (Abstract.) Referring to the well-known difference in taste and other physical properties between the bile as it immediately proceeds from the liver and the same fluid after it has been retained for a time in the gallbladder, the author observes, that the nature of this difference and the agency by which it is effected, are questions which have not yet met with the attention they deserve, and that he had accordingly been led to make them the subject of experimental inquiry. As, however, it is only on rare occasions that the hepatic bile can be procured in quantity sufficient for chemical experiment, and then only at the risk of its being altered by pathological conditions of the secreting organ, the author considers that, however clearly individual facts on the subject may be demonstrated, any deductions made therefrom must be referred to the lower department of probable evidence; and it is with this reservation that he lays his conclusions before the Royal Society, whilst, at the same time, he believes that, so far as the nature of the case admits, he has been able to elicit a new fact respecting the mucous membrane of the gallbladder, which may lead to the better comprehension of the functions of mucous membranes generally. Assuming, in the first place, that the change in properties which the bile undergoes in the gall-bladder is brought about either by the mucous secretion of that reservoir, or by some operation exerted by its internal membrane, it is observed, with respect to the action of the mucus, Ist, that when left in the gall-bladder in contact with the cystic bile, it is capable of subverting the composition of that fluid. 2nd, That this change is much accelerated by even a moderately elevated temperature. 3rd, That when the contents of the gall-bladder are evaporated to a syrupy consistence, the bile, at first VOL. VIII. N |