Also and = 2 +1 Now P remains unchanged when u is written for μ. Hence +1 the summation extending to all positive integral values of m, n, except m = n. Let now +1 +1 Then PPudu-P_P_dμ +["$dμ = ["•dμ. -1 Now (Ferrers's Spherical Harmonics, § 24), +1 Hence = [ødu vanishes unless n = m + 1 or m− 1. Its values in Since μ P, the value of the integral may also be found by sub = = C C m : 2m (2m −3) Am-14m+2m (2m+3) B-1B1+4A which determines X in terms of the A, B. 9. It remains to shew how can be determined in a series of Zonal Harmonics. In doing this we will consider the fluid-motion due to any number of spheres, lying in a line, either at rest, or pulsating, or moving along the line in any manner. This is a slightly more restricted case than is considered above, but the method will apply to that as well. Take any portion of an image within the sphere B under consideration, say μ at a distance P from the centre of B. The potential due to this is therefore the part of B, due to this is p/b and the summation extending to all the sources and sinks within B. So also the summation extending to all without B. In the particular case we are considering, the sources and sinks will be arranged in systems of mass-images, each consisting of a source, and a line sink of constant density, and of magnitude equal to the source. The part of B, depending on the mass-image whose source is μn is VOL. IV. PT. I. except for μ, from which the portion belonging to B, is zero for every i except 0, and Let A, A, B, B" be the parts of A, B, depending on the pulsations of A, B respectively. Then + is the image of vand V1 = a2v, By means of formulæ given in a paper in the Transactions of the Royal Society, 1880, Pt. II., p. 465, the values of p, σ, μ, v can be expressed in terms of the radii of the spheres and their distance. 10. If as in the former paper we neglect that part of the force which varies as a greater inverse power of the distance than the second, it is clear that we do not require to calculate higher terms than A1, B1; and to this degree of approximation it is easily found that, writing now r for the distance of the centres, and it is seen that the parts depending on V have disappeared. In the former paper the forces must be doubled throughout. This arises from an error on p. 280, 1. 10, where 47 must be read for 27. In consequence of this on page 286 read n/2 for n and 1/50/2 or 014 for 1/50. When this correction is made the expression for the force agrees with the above. (3) On the death-struggle of a muscular fibre and the chemical and physical changes which accompany it: together with an explanation of the phenomena of shivering and rigor mortis. By Professor LATHAM. (4) On a sundial of a peculiar form, said to have been reconstructed by Lalande, at Bourg-en-Bresse, in France. By Dr J. B. PEARSON, D.D., Fellow of Emmanuel College. After the termination of the ordinary proceedings, a Special General Meeting, of which due notice in compliance with Section x. of the Bye-laws had been given, was held; at which it was moved by Professor Babington, and seconded by Professor Stokes, that for No. 2, Section VI. of the Bye-laws, the following be substituted: "The President shall take the chair at 3 P.M. and shall quit it before 5 P.M." The motion was carried unanimously. December 6, 1880. PROFESSOR NEWTON, PRESIDENT, IN THE CHAIR. The following communications were made to the Society: (1) On the various notations adopted for expressing the common propositions of Logic. By JOHN VENN, M.A., Fellow of Gonville and Caius College. Most logicians must be well aware of the general fact of the perplexing variety of symbolic forms which have been proposed from time to time by various writers, but probably few persons have any adequate conception of the extent to which this license of invention has been carried. I have therefore thought it well to put together into one list the principal forms, so far as I have observed them, in which one and the same proposition has thus been expressed. For this purpose the Universal Negative has been selected, as being about the simplest and least ambiguous of all forms of statement. This arrangement has not been drawn up with a mere wish to make a collection. Almost every one of these forms, it must be remembered, has been made the instrument of a more or less systematic exposition of the subject. In so far, therefore, as the notation is not entirely arbitrary-which it is in very few instances we shall find it instructive to compare the different aspects of the same operation to which they respectively direct attention. For convenience of reference and comparison |