but about 2', and that in the direction opposite to what is found for an ordinary surface. This 2' can hardly be other than real, for it has been recovered several times after complete resetting of all the apparatus. In any case the ellipticity here presenting itself is exceedingly small. We have x = +tan 30° tan 1'=+00017. The intensity of the light reflected from water at the polarising angle, measured by 2, is not more than about of that found by Jamin. Alcohol is not nearly so dependent as water upon the methods for freeing its surface from contamination; but, on the other hand, I was unable to apply these methods so completely. The value obtained was K = + 00035. A strong brine, cleansed like the water, gave K = -'00042. About the same value applies to a saturated solution of camphor, while for oleate of soda the value was 002. For petroleum again = +0010. It is impossible to feel confidence that these small values really express properties of the liquids whose names are attached to them. What is certain is that, in a large number of cases, the ellipticity is very much less than has hitherto been supposed, and it is not improbable that even the residual ellipticity may be due to contamination, or, if not to contamination properly so-called, to insufficient abruptness in the transition from the one medium to the other. SATURDAY, AUGUST 22. The following Reports and Papers were read:- DEPARTMENT I.-PHYSICS. 1. Sixth Report of the Committee on Electrolysis.-See Reports, p. 122. 2. Interim Report on the Present State of our Knowledge in Electrolysis and Electro-Chemistry. Mr. W. N. Shaw was not able to present a Report this year. 3. Electrolytic Problems. By ROBERT L. MOND. The author establishes the complete analogy between electric conduction through electrolytes and what may be called metallic conduction. He assumes with Wiedemann that the better conducting molecules in the electrolyte form chains, while the worse conducting molecules form dielectric tubes surrounding them. According to Clerk Maxwell's theory, electric energy is transmitted through the dielectric along conductors. The author assumes that this transmission is accompanied by molecular dissociation in the dielectric tubes surrounding each electrolytic chain. With these assumptions he explains the chief electrical and chemical effects produced during electrolysis. The author gives an account of experiments he has made to test the validity of the above views. 4. On Clausius' Theory of Electrolytic Conduction, and on some Secret Evidence for the Dissociation Theory of Electrolysis. By J. BROWN. See the Report of the Committee on Electrolysis, p. 122. 5. Report of the Committee on the Phenomena accompanying the 6. On the Electrification of Needle Points in Air.1 The author measures the strength of the electrostatic field at the surface of a needle point by the mechanical force exerted by the field upon the needle parallel to its axis; and justifies experimentally the formula where ƒ is the field strength at a point of radius of curvature r, and P the mechanical pull upon it. Values of ƒ at the instant of discharge in air are given for air pressures, varying from 10 cm. to 76 cm. of mercury; the measurements having been made on needle points, for which the values of r lie between 7 × 10-4 cm. and 6 x 10-2 cm. It is shown that for radii less than about 10-2 cm. the product fx 708 is fairly constant; its value at 76 cm. mercury pressure being 16.5. In the light of these results the possible ways are discussed in which resistance to discharge may arise at a point. The conclusion is arrived at that the resistance at a clean point is due to the formation of Grotthuss chains of the air molecules surrounding the point; and it is shown that, on this view, the charges carried by the gas atoms are probably of the same order of magnitude as those carried by the same atoms in electrolytes. The variations of f with air pressure are then referred to, and are shown to be in accordance with the Grotthuss chain hypothesis so far as they go. 7. On the Measurement of Liquid Resistances. By J. SWINBURNE. To avoid errors due to variations of resistance or polarisation at the electrodes, the fall of potential over a portion of the electrolyte is measured. Siphon tubes are arranged to connect the feeling points with vessels containing non-polarisable electrodes in a suitable electrolyte. Various ways of arranging the apparatus are described. 8. The Surface-Tension of Ether and Alcohol at Different Temperatures. By Professor WILLIAM RAMSAY, Ph.D., F.R.S. Measurements of the ascent of these liquids in a calibrated capillary tube were made at temperatures varying from that of the atmosphere to within a short distance of the critical point. These measurements, combined with determinations of the angle of contact of the meniscus of the liquid with the walls of a containing narrow-bore tube, and also with a knowledge of the densities of the liquid and the vapour, give data for calculating the surface-tension. The results go to prove that surface-tension is not a linear function of temperature. It is apparently related to the heat of the vaporisation of the liquid in a somewhat simple manner. The angle of contact of the liquid with the tube walls varies in a remarkable manner with the temperature. While, at temperatures for ether up to 160°, the angle of contact is a small and a gradually decreasing quantity, at that temperature it is zero: with rise of temperature above 160°, the angle of contact increases slowly at first, rapidly as the temperature approaches the critical, until at the critical point it is a right angle. It is remarkable that above 160° no bubble will stick in the tube, but ascends to the top; whereas below that temperature a bubble will remain in ' Printed in extenso in the Phil. Mag. September 1891. one position, until it is compressed to such an extent that its form becomes lenticular, the edges of the lens being just in contact with the sides of the tube when it commences to ascend. There appears, therefore, to be a special temperature for each liquid, at which the angle of contact of its meniscus with the walls of the containing vessel is zero. DEPARTMENT II.--MATHEMATICS. 1. Interim Report of the Committee on Mathematical Functions. 2. Interim Report of the Committee on the Pellian Equation Tables. See Reports, p. 160. 3. On Periodic Motion of a Finite Conservative System.1 4. On a Geometrical Illustration of a Dynamical Theorem. It was observed in this paper that a dynamical system when moving in any way could be constrained to adhere to the same motion, so that every element should continue to twist about the same screw as it was twisting about at the moment. The forces to be applied for this purpose could be simply expressed, and a geometrical construction was given in the particular case of a rigid body, which was possessed of three degrees of freedom. It was shown that the screws about which a body so restricted could twist might be represented by points in a plane made on two ellipses, one representing the screws about which the body could twist with zero kinetic energy, the other representing the screws of zero pitch. It was then shown that two homographic systems of points could be constructed such that if any point P be joined to its correspondent Q, then the hole of the ray with regard to the pitch ellipse represents the screw on which a wrench could be placed which should just steady the motion. The pole of the same ray with regard to the kinetic ellipse gives the acceleration of the body if permitted to pursue its movement without interference. A complete account of the investigation will shortly appear in the publications of the Royal Irish Academy. 5. On the Transformation of a Differential Resolvent. If there be two algebraic equations such that they can be changed, the one into the other, by assuming, without loss of generality, certain relations among their variables; and if the differential resolvent of one of these equations is known, how can we pass directly to the differential resolvent of the other, without having recourse to a separate and independent calculation? That is the question I propose to consider in the present paper. Nearly thirty years ago, when seeking to determine the form of the differential resolvents of two trinomial algebraic equations connected in the manner above described, I endeavoured to effect a passage from one differential resolvent to another by a simple transformation, but was stopped by what seemed to me at the time to be an anomalous result. Fortunately the result was placed upon record for future discussion; it will be found in Art. 13 of a paper read before the Literary and Philosophical Society of Manchester (November 4, 1862), 1 Printed in extenso in Phil. Mag. October 1891. and printed in the second volume of the third series of the Society's Memoirs, pp. 232-245. A few weeks ago on re-studying this result, I succeeded in clearing up the supposed anomaly, and in converting one of the differential resolvents into the other. I will here indicate briefly the method employed, as it appears to admit of general application. are The differential resolvents of the equations y" ―ny + (n-1)x = 0 n"-[(n-1)D]"-1y-(n-1) (nD-n-1)[nD-2]n-2xy=[n-1]-1x. respectively, where D = x and the usual factorial notation d dx' [0]=(0)(0-1)(0-2)... (0-a+1) (8') is followed. The question is, how to pass from (a) to (B'); or, in other words, given the differential resolvent of (a) to find that of (3). The following method is effective. These substitutions being made in the resolvent (a') we are led to (n' + 1)n' + 1[— (n' + 1)D′+ 1]−('+1) y − n'2 [ − (n ́D + 1)] - (n'+1x'y' = 0. (7) the result given in the paper above cited. Now, observing that, in general, so that (7) may be written in the form (n' + 1)"'+1[n'D']' + 1y' — n'2[(n' + 1)D'-2]"'+1x'y' =0. • (72) which contains the common factor (D-1), and therefore admits of a first integration. Operating with (D-1)-1, and determining the constant by summing for the (n+1) roots of y'(Ey'n' + 1), we obtain the differential resolvent of (y), namely, - (n' + 1)"'[n'D']"'y' — n'(n' + 1D' — n' + 2)[(n' + 1)D′ — 2]"'—1x'y′ = [n']"'x'. (Y'1⁄2) Dropping accents and writing n-1 for n, we are conducted finally to the equation (B'), the differential resolvent of (B). 6. On the Transformations used in connection with the duality of Differential Equations. By E. B. ELLIOTT, F.R.S. The Monge-Chasles-De Morgan reciprocal transformation of partial differential expressions x' = p, y' =q, z' = px + qy − z, x = p', y=q', z=p'x' + q'y' — z', is readily carried beyond the second order of derivatives by noticing that what is required is to express the derivatives of p, q with regard to p', q' in terms of those of p', q' with regard to p, q. Now a theory of the reversion of partial derivatives of two variables with regard to two others by interchange of the dependent and independent pairs has been developed. The analogous but simpler reciprocal transformation of ordinary differential expressions x'=p, y'-px-y or x = p', y=p'x' -y' amounts only to the interchange of dependent and independent variables in derivatives of p with regard to p'; and a quite complete theory of such a reversion is at our disposal. One consequence is that any reciprocant gives us on replacing a self-reciprocal expression, i.e. the criterion of dp d'p by day day dp' dp a family of curves whose polar reciprocals with regard to the parabola a2 = 2y constitute the same family. 7. Note on a Method of Research for Invariants. This note was of the nature of an inquiry as to whether adequate use had been made of methods of direct determination of invariants of a binary form in terms of its co-efficients when deprived of its second term. The invariants of are as shown by Cayley those functions of a, c, d, dx n-3 + ... whose degree i and weight w satisfy in2w, and which are annihilated by the differential operator 8. On Liquid Jets under Gravity. By Rev. H. J. SHARPE, M.A. The motion, which is in two dimensions, is supposed to be symmetrical with regard to a'Or, which is the axis of the vessel and jet. BEF is the semi-outline of the vessel, FJ of the jet. AF is the semi-orifice which is small compared with the dimensions of the vessel and the depth of the liquid. Gravity acts parallel to a'Ox. OF is the surface of the liquid, which is maintained steady. AF is supposed to be so small that it may be considered either as the arc of a circle with centre O in the surface of the liquid, or as a small straight line perpendicular to Or. For simplicity we shall take OA the radius of the circle (or the depth of the liquid) as unity. If g be the acceleration of gravity referred to this unit, it will be convenient to put a2 for 2g. We shall take O as the origin of Cartesian and polar coordinates x, y, r, 0 and we shall put a' for (x-1). Let x and be the stream functions on the right and left respectively of AF and let u, v be the velocities parallel to Ox, Oy. Further let AF="/р where p is a large number. |