where a, b...... are arbitrary constants to be afterwards determined by the conditions Sc1, SC.......... This result is not new; it was in fact known to Euler, though he admitted that his own demonstration was not complete. I have never yet been able to find in other works a sufficiently rigorous demonstration of this method. On the Dynamical Theory of Gases. By Prof. J. C. MAXWELL. The phenomena of the expansion of gases by heat, and their compression by pressure, have been explained by Joule, Claussens, Herapath, &c., by the theory of their particles being in a state of rapid motion, the velocity depending on the temperature. These particles must not only strike against the sides of the vessel, but against each other, and the calculation of their motions is therefore complicated. The author has established the following results:-1. The velocities of the particles are not uniform, but vary so that they deviate from the mean value by a law well known in the "method of least squares." 2. Two different sets of particles will distribute their velocities, so that their vires vivæ will be equal; and this leads to the chemical law, that the equivalents of gases are proportional to their specific gravities. 3. From Prof. Stokes's experiments on friction in air, it appears that the distance travelled by a particle between consecutive collisions is about th of an inch, the mean velocity being about 1505 feet per second; and therefore each particle makes 8,077,200,000 collisions per second. 4. The laws of the diffusion of gases, as established by the Master of the Mint, are deduced from this theory, and the absolute rate of diffusion through an opening can be calculated. The author intends to apply his mathematical methods to the explanation on this hypothesis of the propagation of sound, and expects some light on the mysterious question of the absolute number of such particles in a given mass. 1 447,000 Supplement to Newton's Method of resolving Equations. This was a mathematical paper, showing a method of greatly shortening and facilitating the finding of the roots of equations of a high order by the method of limits. Note on the Propagation of Waves. This communication aimed at introducing less imperfect geometrical conceptions into the study of wave propagation, than those commonly applied. Each element of the front of a wave has been usually taken as the origin of a spherical disturbance, and the subsequent position of the wave simply regarded as the envelope of all such shells. This mode of treatment has the disadvantage of so imperfectly representing the phenomena, that it leads to great embarrassments. Thus it leaves the direction in which waves are propagated enveloped in great mystery, and most of the methods which have been suggested by geometers for removing the obscurity have failed to be satisfactory. The difficulty at once vanishes if we fix our attention in the first instance on the element whose disturbance at a given moment we wish to determine, and consider, along with its previous condition, all the influences which reach it at that moment. A spherical shell described round this disturbed element as centre, will in general (if the medium be homogeneous, &c.) pass through points from which the influence had started simultaneously; and if the entire series of such shells be considered, as well as the time at which the influence from each must have been thrown off to reach the common centre at the same moment, it will be easily seen, that, roughly speaking, the parts of the medium behind the disturbed centre were to a considerable distance in the same or nearly the same phase when they contributed to its disturbance, whereas those parts in front of it were in rapidly succeeding phases. From this it follows that the influences arriving from behind will have a great preponderating resultant in one direction, while those arriving from the parts in front will almost cancel one another. This clears up at once the maintenance of the onward propagation of an undulation. The points of the medium, which were in strictly the same phase as the disturbed element when they transmitted their influence, lie in general (on the hypothesis of homogeneity, &c.) on a surface of revolution round the wave-normal, and passing through the disturbed point. This surface of revolution (which, if we make the simplest hypotheses, becomes a right cone) is of importance in the theory of waves. If the medium be such that the disturbing influence is but little enfeebled by distance, this cone will obviously be of small angle, and therefore nearly coincide with the backward part of the wave-normal. In such a medium waves will therefore spread but little laterally. A constitution of this kind probably contributes materially to the rectilinear propagation of light, and explains a phenomenon which shows that the common account does not universally hold, viz. the known fact that sound in water bends with less facility round obstacles than sound in air, although the waves constituting it are longer. It is necessary to form a clear conception of what is to be understood by the influence contributed by an element of the medium. The parts beyond one of the spherical shells produce an effect on the central disturbance. This effect is modified by the particular condition in which that shell was at some time previous to the central disturbance. It is this modification which is to be regarded as the influence of that shell; and so of the rest. The resultant is therefore to be obtained by integrating from without inwards. After the conditions which must be attended to when the influence is transmitted from each origin of disturbance with unequal speed in different directions, or is not at a given moment limited to a surface, &c., were referred to, some applications of the method to familiar phenomena which do not admit of easy explanation by the usual methods, were given. On the Relations of a Circle inscribed in a Square. By J. SMITH. On the Angles of Dock-Gates and the Cells of Bees. By C. M. WILLICH. The author showed by trisection of the cube along different planes, the production of various solids, and the relation of these to natural forms known in crystallography, to the bee's cell, and to the theoretical meeting angle of dock-gates (109° 28′ 16′′). Thus a rhomboidal dodecahedron is composed of four rhombohedra. The bee's cell may be imitated by an elongated dodecahedron composed of seven rhombohedra. LIGHT, HEAT, ELECTRICITY, MAGNETISM. On a New Species of Double Refraction. In 1813 Sir David Brewster discovered that when a ray of light is transmitted obliquely through a bundle of glass plates it is completely polarized; but he at the same time noticed that this beam is accompanied with other rays, sometimes nebulous, and sometimes in separate distinct images (depending on the polish and parallelism of the glass), but polarized in an opposite plane. This fact was overlooked by Arago and Herschel in their subsequent researches on the same subject, and was not further pursued by Sir David Brewster at the time. In recently examining, however, several hundred films of decomposed glass of extreme thinness, on which the polish and parallelism of the surfaces enabled him to resume the study of the compound beam, he obtained the following results : 1. When a beam of polarized light is incident obliquely upon a pile of thin and homogeneous uncrystallized films, and subsequently analysed, the transmitted light will exhibit the phenomena of negative uniaxal crystals, that is, it will consist of two * See Phil. Trans. 1814, p. 225-230. oppositely polarized pencils, which produce by interference all the colours exhibited by such crystals under similar circumstances. 2. The two oppositely polarized pencils are, first, the pencil polarized by refraction at each surface; and secondly, the pencil, or rather the fasciculus of pencils reflected from the surfaces of each film, and returned into the transmitted beam. As these phenomena are exactly the same as those produced by double refraction, the author did not hesitate to call the result a new species of double refraction, or a new process in which the phenomena of double refraction are produced. On the Decomposed Glass found at Nineveh and other places. The author described the general appearance of glass in an extreme state of decomposition, when the decomposed part was so rotten as to break easily between the fingers, a piece of undecomposed glass being generally found in the middle of the plate. He then explained how, in other specimens, the decomposition took place around one, two, or more points, forming hemispherical cups, which exhibit the black cross and the tints of polarized light produced by the interference of the reflected with the transmitted pencils. In illustration of this decomposition, he showed to the Meeting three specimens, in one of which there was no colour, but which consisted of innumerable circular cavities with the black cross, these cavities giving it the appearance of ground-glass. In another specimen the film was specular and of great beauty, showing the complementary colours by reflected and transmitted light. In a third variety the films were filled with circular cavities exhibiting the most beautiful colours, both in common and polarized light. Various other remarkable properties of these films were described by the author. On the Submergence of Telegraph Cables. By H. Cox. On the Stratified Electrical Discharge, as affected by a Moveable Glass Ball. By J. P. GASSIOT, F.R.S. If the discharges from an induction coil, when taken in a good carbonic acid vacuum tube, are examined with care, it will be seen that the stratifications nearer the negative terminal are remarkably clear and defined, oftentimes showing clearly separated cloud-like luminosities, but gradually becoming indistinct and intermingled with each other towards the positive terminal wire. This difference in the character of the stratified discharge becomes more perceptible to a certain extent as the vacuum improves; for when the stratifications are close and narrow, they are regularly diffused throughout the entire length of the luminous discharge. In a tube 18 inches long and 1 inch wide, I inserted a small bead of uranium glass about of an inch in diameter. Transparent uranium glass, Professor Stokes has shown has the property of becoming opake by the electric light, and this is very beautifully shown in these tubes, but more particularly when the negative discharge is made to impinge on the bead. If during the discharges the tube is inclined so as to permit the bead to roll down, the discharges will give the appearance as if a distinct row of separated beads were present; this appearance arises from the number of discharges which take place during the rotation, each discharge separately and distinctly illuminating the bead. The peculiar phenomenon which I, however, desire to bring before the notice of the Section is one which I only very recently noticed. I have already stated that the stratifications near the positive wire are indistinct; but if the glass bead is placed near the positive wire and then allowed slowly to descend towards the negative, the stratifications at the positive are at first as clearly defined near that terminal as at the negative, and as the bead rolls gently down, they have the appearance of following the bead and issuing one after the other from the positive wire, until the bead reaches to within a few inches of the negative, when this action gradually ceases. If the tube is now inclined so as to allow the glass bead to return in the contrary direction, the stratifications appear to recede, becoming more and more clearly defined, until the bead passes the positive terminal wire, when the entire discharge returns to its normal state. On the Relation between Refractive Index and Volume among Liquids. By the Rev. T. P. DALE and J. H. GLADSTONE, Ph.D., F.R.S. The authors referred to a previous paper, in which they had shown, among other things, that the sensitiveness of a substance is not directly proportional to the change of density produced by an alteration of temperature. The theoretical formulæ relating to the dispersion of light afford little assistance in determining what this relation is, but a series of careful observations had been made with a view of arriving at some empirical formula. It was found that the product of the volume, reckoned as 1000 at the boiling-point, and the refractive index for the line A of the prismatic spectrum less unity, gave numbers which were nearly constant. In the case of water, alcohol, pure wood-spirit, and bisulphide of carbon, however, the volume increases a little faster in proportion than the refractive index less unity diminishes, while with ether the reverse is the case. The regularity of the numbers shows that this is not due to errors of experim nt. The authors propose examining the subject more closely. On the Theory of Light. By G. F. HARRINGTON. Notice of Experiments on the Heat developed by Friction in Air. The research which Professor Thomson and myself have undertaken on the thermal effects of fluids in motion, naturally led us to examine the thermal phenomena experienced by a body in rapid motion through the air. The experiments which we first made for this purpose were of a very simple kind. We attached a string to the stem of a sensible thermometer, and whirled it alternately slowly and rapidly. In this way we uniformly obtained a slight effect; there was a higher temperature observed immediately after rapid, than after slow whirling. A thermo-electric junction rapidly whirled also gave us an appreciable thermal effect, indicated by the deflection of the needle of a galvanometer. Afterwards a more accurate set of experiments was made by us; using a lathe, to the spindle of which an arm was attached carrying one of Professor Thomson's delicate ether or chloroform thermometers. The thermometers employed were so extremely sensitive that each division of their scales had a value of not more than of a degree Centigrade. The great value of Professor Thomson's thermometers in the whirling experiments, was further enhanced by the light specific gravity of ether comparatively with mercury: the pressure produced by centrifugal force operating on a long column of mercury, would have probably broken a mercurial thermometer whirled at high velocity. The results arrived at by Professor Thomson and myself were as follow :— 1st. The rise of temperature in the whirled thermometer was, except at very slow velocities, proportional to the square of the velocity. 2nd. The velocity at which the bulb had to travel in order that its temperature should be raised 1° Cent. was 182 feet per second. 3rd. At very slow velocities the quantity of thermal effect appeared to be somewhat greater than that due from the square of the velocity calculated from the above datum; and we surmised that this was owing to a sort of fluid friction different from the source of resistance at high velocities. We therefore made several attempts to increase this fluid friction; the most successful result being obtained by wrapping fine wire over the bulbs. By this means we succeeded in obtaining the from a velocity of 30 feet per second, a quantity five or six times as great as that which took place when the naked bulb was revolved at the same velocity. We resumed the whirling experiments last May; and it is owing to the circumstance that it has happened that I have myself been principally engaged in making those which I am about to communicate to the Section, that Professor Thomson has requested me to give an account of this part of our joint labours. Our object was to repeat the former experiments under new circumstances, so as to verify and extend the results already obtained. A very brief outline can only be given in this place, as we intend shortly to incorporate them in a joint paper for the Royal Society, to whose assistance we owe the means of prosecuting the inquiry. The lathe was again used as the whirling apparatus, but instead of the ether thermometer, we whirled thermo-electric junctions of iron and copper wires. We obtained the following results : 1st. The law of the thermal effect was, as with the ether thermometer, proportional to the square of the velocity. 2nd. The rise of temperature was independent of the thickness of the wire which formed the thermo-electric junction which was whirled. This was decided by experiments on wires of various diameters, ranging from to of an inch diameter. The rise of temperature was in any of the wires the same as that obtained with the ether thermometer, the bulb of which was nearly half an inch in diameter. 3rd. The thermal effect appeared likewise to be independent of the shape of the whirled body; little difference happening in whatever direction the wire was placed. 4th. The average result was that the wire was warmed 1° by moving at the velocity of 175 feet per second. The highest velocity obtained was 372 feet per second, which gave a rise of 5o.3, and there was no reason to doubt that the thermal effect would go on continually increasing according to the same law with the velocity. Thus at a mile per second the rise of temperature would be 900o, and at 20 miles per second, which may be taken as the velocity with which meteors strike the atmosphere of the earth, 360,000°. The temperature due to the stoppage of air at the velocity of 143 feet per second is one degree. Hence we may infer that the rise observed in the experiments was that due to the stoppage of air, less a small quantity, of which probably the greater part is owing to loss from radiation. It being also clear that the effect is independent of the density of the air, there remains no doubt whatever as to the real nature of "shooting stars." These are small bodies which come into the earth's atmosphere at velocities of perhaps 20 miles per second. The instant they touch the atmosphere their surfaces are immediately heated far beyond the point of fusion, or even of volatilization, and the consequence is that they are speedily and completely burnt down and reduced to impalpable oxides. It is thus that, by the seemingly insufficient resistance of the atmosphere, Providence secures us effectually from a bombardment which would in all probability speedily destroy all animated nature, with the exception of the fishes, which would be partly, but not altogether, protected by the water in which they swim. The experiments to carry out and verify our previous results on the thermal effects which appear to belong to friction on large surfaces at slow velocities were made as follows:-A disc of zinc or card-board was attached to the revolving axis. An ether thermometer was attached to the disc, the bulb being near the circumference and describing a circle with a radius of about 1 foot. On rotating the disc at the velocity of 1 foot per second, as much as one-thirtieth of a degree of heat was developed. On the Transmission of Electricity through Water. By J. B. LINDSAY. The author has been engaged in experimenting on the subject, and in lecturing on it in Dundee, Glasgow, and other places since 1831. He has succeeded in transmitting signals across the Tay, and other sheets of water, by the aid of the water alone, as a means of joining the stations. His method is to immerse two large plates connected by wires at each side of the sheet of water, and as nearly opposite to each other as possible. The wire on the side from which the message is to be sent is to include the galvanic battery and the commutator or other apparatus for giving the signal. The wire connecting the two plates at the receiving station is to include an induction coil or other apparatus for increasing the intensity and the recording apparatus. The distance between these plates he distinguished by the term "lateral distance." He found that there was always some fractional part of the power from the battery sent across the water. There were four elements on which he found the strength of the transmitted current to depend: first, the battery power; second, the extent of surface of the immersed metal sheets; third, the "lateral distance" of the immersed sheets; and, fourth, in an inverse proportion the transverse distance or distance through the water. As far as his experiments led him to a conclusion, doubling any one of the former three doubled the distance of transmission. If, then, doubling all would increase the intensity of the transmitted current eightfold, he entered into calculations to show that two stations in Britain, one in Cornwall and |