6 equations by the invention of an algebra of non-commutative symbols. To take the simple instance given by Ellis. Noah the father of Shem: ... A son of Noaha son of the father of Shem=Shem. The process is formally identical with the following: Examples of inference lying beyond the domain of the old logic are deserving of much greater attention than they have hitherto received. Professor De Morgan seems to be the only writer who has treated of such examples with any degree of fulness and ability. (See papers on the Syllogism, and on the Logic of Relations, in the Cambridge Philosophical Transactions.) On Complexes of the Second Order By Dr. PLUCKER, F.R.S., of Bonn. Dr. Plücker showed a series of models executed with great accuracy by Mr. Epkens of Bonn, calculated to illustrate his theory of complexes of the second degree. Such complexes are determined analytically by equations of the second degree between the coordinates of right lines in space. of such a complex envelope a curve of the second class, and every point in space is In any plane whatever the lines the centre of a cone of the second order generated by lines of the complex. If a plane revolves round any line within it regarded as an axis, the variable conic therein generates a surface. The same surface is enveloped by a variable cone of the second order, the centre of which moves along the same axis. Surfaces of this description are of the fourth order and the fourth class. The axis is a double line of the surface. The four circumscribed cones whose centres are the four intersections of the double line with the surface, degenerate into systems of two planes, each of which touches the surface along a curve of the second order. In each of four planes passing through the double line, the conic degenerates into two points; these points (singular points of the surface) are the centres of cones formed by tangents to the surface. The poles of the double line, with regard to all conics in planes passing through it, are situated on a right line, through which pass the polar planes of the double line with regard to all circumscribed cones. The surfaces even of the more general description are easily constructed; the models exhibited belong to the special case where the double line is at an infinite distance. In this case the surfaces are formed by curves of the second class in parallel planes, having their centres on a light line. The circumscribed cones circumscribed cylinders. On the Hyperelliptic Functions, Göpel and Weierstrass's Systems. By W. H. L. RUSSELL, A.B., F.R.S. The author of these papers gave an explanation of the methods discovered by Göpel and Rosenhain for the comparison of the hyperelliptic functions. pointing out their enormous complication, he stated that a simpler method had been discovered by Dr. Weierstrass, which he illustrated by showing how Abel's After theorem had been employed by that mathematician in deducing the periods of elliptic and hyperelliptic functions. On a Property of Surfaces of the Second Order. By H. J. S. SMITH, F.R.S. On the large Prime Number calculated by Mr. Barratt Davis. On a Nomenclature for Multiples and Submultiples to render absolute Standards convenient in practice, and on the fundamental Unit of Mass. By G. J. STONEY. By On the Partition of the Cube, and some of the Combinations of its parts. CHARLES M. WILLICH, late Actuary and Secretary to the University Life Assurance Society. A cube may be divided into equal and uniform bodies in various ways. 1st. By lines from the centre to the eight angles of the cube, which will give six four-sided pyramids (B). 2nd. By lines from one of the upper angles of the cube drawn diagonally to the opposite angles, dividing the cube into three equal and uniform solids. Each of these solids being halved, forms a left- and right-handed solid. These six bodies produced, though equal in mass, differ so far in shape, as three may be termed lefthanded and three right-handed, in the same way as the hands of the human body. 3rd. By lines drawn from the centre to four angles of the cube, and continued on each face, will produce four equal and similar bodies (G), each composed of two threesided pyramids united at their base-the one having the same angle as the trihedral roof of the Bee's cell, viz. 109° 28′ 16", the other 90°. These bodies rearranged produce the half of a dodecahedron with rhomboidal faces. 4th. Another division of the cube may be made producing the tetrahedron and octahedron, viz., by diagonal lines from two of the upper angles of the cube, continued on the other faces, will cut off four three-sided pyramids, leaving in the centre a tetrahedron. The four three-sided pyramids cut off may be so arranged as to produce the half of the true octahedron. The four-sided pyramid obtained by the first mode of division being cut into two portions by a diagonal line will produce a body which I have assumed as a unit (A) for the construction of many geometrical and crystalline bodies. The models laid before the Association show some of the forms produced. The rhomboidal cube (J), and the rhomboidal dodecahedron (L) with pyramidal faces (containing in mass one-half of the cube from which it is derived), may be considered interesting; but the various crystalline figures which may be formed by a combination of my unit (A) I cannot even estimate-though probably all geometrical solids and even many, if not all, crystalline bodies may be included, if we use sections of bodies produced by a partition of the cube. It may be observed that the pyramid (B), or one-sixth of the cube, which contains two units, may itself be divided into four bodies by sections parallel to the sides, each of which is one-third of a cube containing one-eighth of the mass of the cube from which it was derived; so that, in fact, we may go on dividing and reproducing bodies of a similar shape, and still retaining the same angles as in the portion from the original cube. How far this subdivision may be carried in nature, or how much further than our powers of vision will reach, I will not venture an opinion. We can imagine that the commencing atoms may be infinitely small when we remember the wonders revealed by the microscope. I entertain a sanguine hope that, should the attention of philosophers be drawn to this subject, the further development may perhaps be the means of throwing some unexpected light as to the shape of an atom. I incline, however, to think that atoms may differ in shape in the three kingdoms of nature-mineral, vegetable, and animal. As to the practical use in education, I am of opinion that the study of geometry would be simplified by the use of models showing the relative value as to the solidity of geometrical bodies, and thus convey knowledge to the youthful mind by means of the eye more readily than by any description, as when convinced by the sight the mind would understand with greater facility. A. The unit or List of models which accompanied the above paper. part of the cube having a side of 1 inch. B. 1st union of two units, forming a low four-sided pyramid of which six make up a cube. C. 2nd union of two units, forming a high four-sided pyramid of which six also equal cube. D. 3rd union of two units, forming right-handed solid, being of cube. F. 5th union of two units, forming part of body G, which is the fourth part of cube. G consists of 3 units, forms one-fourth of cube, and is the body obtained by the third mode of dividing the cube. H. Cube composed of 4 of the above bodies, G. I. Four of the same bodies (G) reversed and rearranged to produce the half of the rhomboidal dodecahedron. Another cube similarly divided and arranged completes the solid. J. Six units or three of C so arranged as to produce the rhomboidal cube-the basis of the hexagonal system. Seven of these bodies build up the bee's cell. K. The cube divided by cutting off four three-sided pyramids, leaving a tetrahedron in the centre. The four three-sided pyramids cut off may be so arranged as to produce the half of the true octahedron. L. Rhomboidal dodecahedron with pyramids (C) on each of the twelve faces. This body contains forty-eight units (Â). M. The remainder of the large cube having a side of 2 inches, consists of fortyeight units (A) so arranged as to show how the rhomboidal dodecahedron (L) can be inserted in the vacant space. ASTRONOMY. Remarks on the Variable Star lately discovered in Corona Borealis. Early in June last the author received a letter from Mr. W. Barker, of the Customs Department, London, Canada West, stating that the remarkable variable star in Corona Borealis, which was seen in Europe on May 13, had been discovered by him on the 4th of that month. He thus describes its variations:-"I first observed it on the 4th of May at 9 P.M., when it was somewhat brighter than Epsilon Coronæ ; it rapidly increased until the 10th, when it was fully as bright as Alphacca (Alpha Coronæ); it was at its maximum. On the 14th it had decreased to the third magnitude, on the 18th to the fifth. On the 19th I could just discern it, and on the 20th I could see it no longer with unaided vision. On the 20th I observed it through my telescope (one of Cooke's 5 feet 4 inch object-glasses). With a power of 133, it showed a beautiful clear disk, and was exceedingly brilliant, and had a ruddy tinge. I still see it as a telescopic star; its light about equal to the companion of Polaris." As far as the author was at present informed, Mr. Barker did not make a public announcement of his discovery until the 16th of May, when he communicated a paragraph to the London Free Press,' and forwarded copies of the paper to various astronomers in this country. It runs thus:-" Astronomers will be interested to learn that a new star has made its appearance in the constellation of Corona Borealis. It is of the third magnitude, and is situated about one degree S.E. by E. of Epsilon Coronæ, and three degrees from Pi Ophiuchi, in a direct line between the two. It also forms the apex of an equilateral triangle with Beta and Zeta Herculis. Hour of observation, 9 P.M., 14th May, at London, C.W." It will be remarked that in this communication no reference is made to any observation of the star previous to the 14th of May, probably because Mr. Barker merely intended his notice to refer to its appearance at the date of his letter. But these observations are of historical and scientific value; and the author has not failed to press for any further particulars or corroborative facts which it may be in Mr. Barker's power to furnish. Several European astronomers, ignorant of Mr. Barker's observations, have conjectured that the star must have burst forth with astonishing suddenness. Mr. Schmidt, of Athens, a practised observer, thought it could not have been so bright as a star of the fifth magnitude on the 12th of May, early in the evening, or he must have perceived it; and M. Courbisse, at La Rochelle, was convinced it was invisible to the naked eye on the 11th; yet at this date it must have shown, according to Mr. Barker's observations, as a star of the second magnitude. This is by no means a solitary instance in proof of the little value which attaches in many cases of a similar kind to merely negative evidence. In his own astronomical practice the author had met with startling instances, and striking ones may be found in the history of these phenomena of variable stars. Tycho Brahé thought the celebrated new star of 1572, which he detected on returning home from his laboratory, and which was then shining as a star of the first magnitude, could not have been visible an hour or so previously, and yet, keen observer as he was, he is well known to have been preceded by several days in the discovery of that wonderful object. Astronomers generally, however, may not be disposed to attach so little weight to negative evidence in a case of this kind, as from his own experience Mr. Hind was inclined to do, and it will be most desirable to possess every particular relating to Mr. Barker's observations between the 4th and 14th of May, which it may be in his power to furnish. Mr. Barker thinks he saw this star one or two years earlier, when the constellation was in the S.E., about 9 P.M., and Sir John Herschel announces his having recorded a star in this very position in one of his revisions of the heavens. The apparition of this star will be memorable as having afforded an opportunity of applying the spectrum-analysis to one of this class of objects. The valuable and highly interesting observations by Mr. W. Huggins and others are the results. LIGHT. Optics of Photography.-On a New Process for equalizing the Definition of all the Planes of a Solid Figure represented in a Photographic Picture. Means of producing Harmonious and Artistic Portraits. By A. CLAUDET, F.R.S. [This paper was published in the Philosophical Magazine for September 1866.] On a New Geometrical Theorem relative to the Theory of Reflexion and Refraction of Polarized Light (Isotropic Media). By M. A. CORNU. The direction of the luminous vibration relatively to the plane of polarization of a ray has not been yet stated in a way which is quite incontestable. Fresnel, in his admirable memoir On the Mechanical Theory of the Reflexion and Refraction of Polarized Light,' concludes that the vibration is perpendicular to the plane of polarization. McCullagh and Neumann have arrived at the same formulæ, but by supposing, on the contrary, that the vibration is within the plane of polarization. It seems that no middle term can exist between these two theories, and that the three rays have necessarily their vibration in the identical position compared with their respective plane of polarization. However, there is a third method, or, in other words, a third theory, extremely simple, the author would not say extremely plausible,-which will lead us to the opinion of Fresnel respecting the refracted ray, and to the opinion of McCullagh respecting both the others. The only principle to be admitted, besides the exact transversality of the vibrations, is the following-the refracted vibration is perpendicular to the incident and reflected vibrations. We have, indeed, no theoretical ground for admitting, à priori, this principle; but if the consequence of it agree with the results of the other theories, it deserves to attract the attention of theorists in optics, and, in fact, it will constitute a new theorem. With the help of this principle, it is easy to determine the position of the reflected and refracted vibrations, if the position of the incident vibration is given. The resulting formula is in which a, B, y are the angles of the incident, reflected, and refracted vibrations with the plane of incidence, i and r the angles of incidence and refraction. Seeing that the vibrations are besides transversal, the above formula determines them completely. But if this theory is exact, that formula is nothing else than the analytical translation of the law of the rotation of the planes of polarization of the three rays a law first stated by Fresnel, and which, according to the same rotations, may be written = cot a =cot y, a, B, y being the angle of the vibration with the plane of incidence. McCullagh It is obvious that our formula agrees with the formula of Fresnel for the refracted ray, and with the formula of McCullagh according to the incident and reflected rays. It is easy to conclude, from this theory, that under the normal incidence the luminous vibration rotates a right angle when the ray penetratos into the second medium. It would be interesting to look for a direct verification of that conclusion; but it seems difficult to realize an experiment in which the surfaces limiting the medium do not produce an even number of those rotations, so that the vibration does not come again to its first direction. The author could have stated this property of polarized light under a more modest form, that is to say, as a simple corollary of known theorems; but he fancied that it was more useful, in the actual state of optics, to state it as a new theory, in order to show, first, that the geometrical simplicity of the principles does not constitute the most plausible theories: thus it is prudent to conclude that the greater geometrical simplicity of the McCullagh theory is no sufficient ground for rejecting the theory of Fresnel, though more complicated. Besides, the proposal of a new principle, very little obvious, à priori, is a good occasion to remember the feeble degree of evidence for the principles used in the other theories. After a further examination, it will appear that it is neither more nor less difficult to admit that the refracted vibration is perpendicular to both the others, than to admit, for the density of the luminous ether, the theories of Fresnel or McCullagh. μ μ On Dispersion-equivalents. By Dr. J. H. GLADSTONE and Rev. T. P. DALE. The refractive index of a substance minus unity, divided by its density, is termed its "specific refractive energy," and the product of this and its atomic weight, or Р is termed its "refraction-equivalent." But the refractive index u differs d' according to the part of the spectrum observed. As the fixed lines A and I are the extremes in the two directions which can be measured under ordinary circumstances, μ-μA has been taken as the measure of dispersion; and in a previous μη μ paper the authors had called the "specific dispersion." Hence the dif ference between P d μ-1 d the "Dispersion-equivalent;" and as P is little affected by the manner in which the substance is combined with other bodies, it becomes a matter of interest to inquire whether the same holds true with respect to P "„—”a, d It has been abundantly shown that the refraction-equivalent of the combination CHI, is 76: its dispersion-equivalent, as determined from eight different substances or series belonging to the great vinic group of organic compounds, varies only within the limits of 0:32 and 0:38, the mean being 0:35. But when we turn to the benzole group its dispersion-equivalent is found to be 062, and in the pyridine group 0.58. The determinations of the dispersion-equivalents of chlorine, bromine, and iodino also differ, when they are made from different groups. Phosphorus is an extremely dispersive body, and when in a liquid condition gives for P the high number 2.9, though the value of P for this element is low. d Spectroscope de poche. Par Dr. J. JANSSEN. Cet instrument est à vision directe, et forme une très-petite lunette. Le redresse |