UNIVERSITY CALIFORNIA NOTICES AND ABSTRACTS OF MISCELLANEOUS COMMUNICATIONS TO THE SECTIONS. MATHEMATICS AND PHYSICS. MATHEMATICS. On the Solution of Cubic and Biquadratic Equations. On an Extension of Quaternions. By Sir W. R. HAMILTON, LL.D., M.R.I.A. The author showed how, by a slight change in the formulæ of quaternions which he had already published, and a slightly different interpretation of the sector and versor, they became applicable to the investigation of the properties of the spheroid. The Abbé Moigno presented to the Section a new Arithmometer, or Calculating Machine, by M. T. De Colmar; and as the Abbé spoke English with difficulty, he requested Prof. Wilson to explain the machine to the Section. The machine, which was very beautifully executed, consisted of an oblong box, about 30 inches long by 6 inches wide. On the face, the machine was furnished with a handle to turn round a number of small holes, at which the digits of the common arithmetic scale, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, made their appearance as the machine worked, and which finally gave the answer. In this machine they were eight in number, but they might be extended to any number. To each of these was an index to be set to the required digit, engraved on a small attached vertical scale, and a small ivory ball to be moved along its scale according to certain simple rules, as the operation to be conducted by the machine varied from addition to multiplication, &c. Upon drawing out the sliding bottom of the machine, the machinery was exposed to view. This, though simple, could not be intelligibly explained without the machine or diagrams. The chief part of it consisted of eight cylinders so arranged, that, as they turned, the digits, enamelled on a circle at their upper parts, came in succession to the holes in the face; while by a number of indentations arranged spirally round them the digit to which the index was set would be stopped at the hole on the face at the digit corresponding to that at which the index was set; while by a set of pinions a connexion was given to them something similar to that in the common bank-note machine, so that addition could be performed and the result appear on the face :—thus by 1854. 1 turning the handle once, the number itself appeared; by turning it twice every digit in it was doubled; and the result appeared above as twice the number originally set, and so on with any multiple of the number so set; then by moving the ivory ball any simple multiple of 10 times, 100 times, 1000 times, the number set could be ob tained and added to those previously obtained, and thus the operation of multiplication performed of any number by any number to the extent the machine could give, in this case up to 99,999,999 or nearly 100,000,000. The Professor then exemplified this, by setting a large number and multiplying it by a number which consisted of three digits. He then explained how the other operations were to be performed,showing that the machine could add, subtract, multiply, divide, raise to an integer power, or extract the square or cube root with precision and rapidity. The price of the machine exhibited was £50. Prof. Wilson then, on the part of the Abbé Moigno, presented and explained Babinet's Homalographic Maps. In the maps on Mercator's projection, although the relative geographical positions were accurately and simply laid down, yet there was a great distortion, particularly in those lands and seas towards and about the Polar regions; and the same remark was more or less applicable to all the ordinary projections used in maps. But in those of M. Babinet, by making the principal meridians in both hemispheres straight lines, and the others, on each side of them, arches of ellipses passing through the Polar points, and their ellipticity varying with their position by a simple law, the exact harmony and proportion of the several parts, land and water, countries and places, on the map were correctly preserved when accurately laid down. M. Moigno presented to the British Association through this Section a map of the world on this construction. LIGHT, HEAT, ELECTRICITY, MAGNETISM. On a New Photometer. By M. BABINET, of Paris. This photometer consisted of a tube, at one end of which was a Nicol's Prism, through which the light to be valued is admitted, the radiant or source of the light being placed at a measured distance. As it passes along the tube the light encounters a bundle of glass plates through which, as it passes, it is polarized by refraction. It then passes on and is received at the eye-pieces; another tube, furnished also at its extremity with a Nicol's prism, also enters the side of this first tube at such a place and at such an angle as that light admitted from a standard source at a fixed distance is reflected to the eye-piece by the same bundle of parallel glass plates, through which the former light is refracted. By turning the Nicol's prisms, exact complementary colours can be had from each source, and where the images of oblong slits, through which the light passes, are made to cross at the eye-piece, the crossed part will be free from either colour when the light to be tried is at exactly the distance which gives the same intensity to the light which enters the instrument as that which comes from the standard. A comparison of the squares of these distances gives the intensity of the light to be valued, in the usual and well-known method. Note on Indices of the Refraction of Transparent Media included between two Parallel Faces, and on a Portable Refractometer, exhibited to the British Association for the Advancement of Science. By Dr. Félix BERNARD, of Bordeaux. The indices of refraction of parallel plates and liquids may be measured by a very simple method founded on the following principles: 1. Every ray that falls perpendicularly upon the surface of a transparent medium with parallel faces traverses it without deviation. The author had already made a communication to the Academy of Sciences relative to the determination of the indices of thick plates, but the apparatus referred to in this Note had not been exhibited before that body. 2. When the incidence is oblique, the ray refracted in this medium emerges from it in a direction parallel to that of the incident ray, and the distance included between these two directions, or the transport, only depends on the angle of incidence, on the index of refraction of the refractive medium, and on the thickness of this medium. From the relation subsisting between these four quantities we may therefore deduce the index of refraction, if the three others be known; the incidence may be chosen at pleasure, the thickness is easily measured, and the transport, the only unknown quantity, is determined by observation. Let a be the angle of incidence, p the angle of refraction, e the thickness of the medium, and n the index of refraction for a ray which traverses the plated the distance sought, then hence sin a n=. tang ae tang p+ sin p This formula will give n, when e being measured and a given, d has been determined experimentally. The apparatus which serves for the determination of d is composed— 1. Of a sight formed by a spider's thread fixed vertically in the interior of a horizontal tube; at one extremity of this tube is a diaphragm, through the aperture of which the light penetrates into the apparatus; a lens placed between the sight and the diaphragin, and capable of motion in the direction of the axis of the tube, allows the light received to be conveniently regulated. 2. Of a horizontal circle for the determination of the incidences; at the centre of this circle there is a support which may be turned by an alidade traversing the limb of the circle. 3. Of a telescope furnished with two very fine wires crossing in the focus of the eyeglass; its optic axis is parallel to that of the tube containing the sight, and it is capable of motion in a direction perpendicular to this axis. The amount of motion is measured and effected by means of a micrometric screw. Mode of operation.-The index of the alidade is placed so as to coincide with the zero of the circle; the telescope is pointed towards the sight, and the lathe is placed in focus; the plate is then fixed vertically against a frame placed on the horizontal tablet of the support, and by turning it on its axis, without disarranging the alidade, the position which it should retain, which is such that the sight may be perceived without deviation, may be found in a few seconds. To determined, the alidade is turned to an angle a, and then to an angle -a moving the telescope so as to point it upon the sight in both positions of the refracted image; the space passed through by the optic axis of the telescope, ascertained by means of the screw, gives the value of d. The very trifling errors which might arise from a slight obliquity of the plate in relation to the direction of the incident rays may be compensated in the following manner :-If the plate be slightly prismatic, it is turned upon its axis 180°, and the distance of the emerging parallel rays is again measured; the average of the values thus obtained will give a very close approximation to the distance sought. I have ascertained positively that a want of parallelism in the plates, even when very sensible by the spherometer, does not affect the exactitude of the observations. The sight may receive either the rays of the solar spectrum, or the light of a strong lamp modified by suitable absorbents, or it may be illuminated by a bundle of white light; in this the average index of refraction of the refractive body is obtained, and this is very often sufficient. The thicknesses may be measured by means of any spherometer, but in this case it is necessary to determine the relation between the lengths of the turns of the screws in the two instruments, which is not attended with any difficulty; it is sufficient to determine the value of d, for a substance of which the relative index to a ray of the spectrum is known; the value of e may then be deduced from the preceding formula. This quantity will usually differ from that furnished by the spherometer, but their relation will always enable the values of e and d to be compared with the same nicety. Nevertheless, in order to avoid this correction, M. J. Duboscq has given a very simple disposition to the micrometric part of the apparatus, so that when the telescope which moves the screw is removed, it readily forms a very delicate sphero meter. By operating in this manner, but with a very imperfect apparatus, I found, for the indices of refraction of the ordinary ray by plates of quartz of 7 to 8 millimetres in thickness, the numbers 1.5441, 1.5473, 1.5496 and 1.5541, which correspond respectively to the rays D, E, F, G. These numbers do not differ from those given by M. Rudberg by more than ths at most. 2 10,000 This process may be advantageously employed in researches relative to the indices of refraction of liquids; the mode of operation is so simple that I may dispense with entering into details upon this subject. Description of a Photometer exhibited to the British Association for the Advancement of Science. By Dr. FÉLIX BERNARD, Bordeaux. The photometer which I have the honour to bring before the Association has been exhibited by M. Arago to the Academy of Sciences, at its meeting of the 9th of May 1853. This instrument, which is principally intended for the study of the transparency of bodies, may be very advantageously employed in a great number of photometric investigations, regarding the illuminating power of luminous bodies, the diffusion of light, and its reflexion from the surfaces of bodies, &c. I shall describe particularly the arrangement which I have adopted for ascertaining the absorption exercised upon light by the action of transparent media. In this case the anterior portion of the instrument presents a tube furnished with two diaphragms, through which the luminous rays, previously rendered parallel by means of a lens, penetrate into the apparatus. At their entrance into the body of the instrument, the rays fall perpendicularly upon the middle of the surface of a double prism, formed by the union of two equal, rectangular, isosceles prisms, of which two faces are in the same plane. The bundle formed by these rays, on meeting the edge of the right dihedral angle of the hypothenuses, is divided into two other equal bundles, which are entirely reflected in opposite directions perpendicularly to that of the incident bundle. At a few centimetres from the axis there are two reflecting prisms, by means of which the rays are reflected in a direction parallel to the axis. Farther on the new bundles traverse some Nicol's prisms which serve to polarize and analyse the light, and by a second arrangement of reflecting prisms symmetrical with the first, these bundles, after passing through the Nicol's prisms, are again turned towards one another and reflected upon the hypothenuses of a second double prism, so as, by their reunion, to reconstitute the primitive bundle. This bundle then traverses a tube forming the ocular portion of the instrument, and arrives at the eye as though it had never undergone any deviation. At one extremity of the ocular tube and close to the surface of emergence of the double reflecting prism, there is a diaphragm, the aperture of which cuts off a circular image upon this surface; this is magnified by a lens. The rotation of the analysers is measured upon two vertical semicircles by means of alidades furnished with verniers. The apparatus being arranged, that is to say, the extinction being complete when the verniers are at zero, the intensity of each half of the circular image perceived by the eye varies in proportion to the square of the sinus of the angle described by the principal section of the analyser; and consequently if one of them be rendered more obscure by the interposition of an absorbent medium in the course of the bundle which corresponds with it, it becomes easy to estimate its intensity, when, by a suitable rotation of the analyser corresponding with the other half, the two parts of the image have been rendered uniform in tint. The artificial light emanating from a strong lamp may be operated with, but before arriving at the apparatus, this light should traverse a system of absorbents arranged in such a manner that the light transmitted may belong to one of the principal tints, almost monochromatic with the solar spectrum. It is better, however, to employ the direct light of the spectrum, by receiving on a screen furnished with a slit, a very fine vertical spectrum formed in the usual manner by rays rendered parallel by means of lens (the principal lines of Fraunhofer should then be seen distinctly); the rays which |