The conditions favourable for the formation of hoar frost occur so abundantly in nature that it is surprising that the forms of natural hoar frost have been until recently unknown. In 1892 one of us found the walls of an ice cave in Iceland coated with some remarkably fine hoar frost crystal, of a shape hitherto unknown, but since then found by us to be the typical and principal form of hoar frost crystals, viz., hollow hexagonal pyramids. It was found that the most favourable conditions for the formation of large crystals are not only moist air at a low temperature. but also a quite undisturbed state of the air. For this reason the finest crystals are formed in caves or, generally speaking, in closed spaces. With regard to the forms of hoar frost, the most typical form, as already men tioned, is a hollow hexagonal pyramid. It is built up upon a small flat hexagonal prism, which springs from a solid wall; round its edges a hexagonal step is formed; round the outer edge of this, again, another larger hexagonal step, and so forth, exactly analogous to the hollow salt hopper crystals of the cubic system. Like the latter, these hollow hexagonal pyramids are the product of the struggle for attraction of material from the surrounding atmosphere; the outer edges, having a wider area to attract material from, grow more than the central portions, which remain uncompleted. We have thus skeleton crystals' formed; the centre remains undeveloped, due to the want of material to 'starvation.' The greater possibilities of attraction are well exemplified by the additional crystal formations at the outer angles of the hexagons, quite similar to the 'hopper' crystals of NaCl. The crystalline forms are exclusively those of the hexagonal flat-topped prism; never has a terminal pyramid nor a hemihedral shape been observed. A great variety of forms of skeleton crystals can be observed under favourable conditions; amongst others, helix-shaped hollow pyramids, analogous to the cubic helices of bismuth; also long solid or helix-shaped hexagonal prisms. The hollow pyramids are built of steps of prismatic rings, invariably with a basal pinacoid face. Often a crystal shows needle-like spikes arranged in decided right angles. This gives the strongest impression of cubic or other rectangular crystals. On careful examination, however, it will always be found that in such a case we have to deal with the incompletely developed rectangular faces of the hexagonal prism, such as we might expect in a skeleton crystal. A series of micro-photographs and sketches will illustrate this. The most favourable conditions for the formation of hoar frost crystals and the best opportunities for studying them are found in the refrigerating chambers as used extensively in Liverpool, and through the kindness of the large shipping firm of Messrs. Nelson an excursion and demonstration will be held there. DEPARTMENT OF PHYSICS. 1. Discussion on the Treatment of Irreversible Processes in Thermodynamics. Opened by J. SWINBURNE, M.Inst. C.E. The following Papers were read : 2. Note on the Rate of Combustion and Explosive Pressure of Cordite. By J. E. PETAVEL. The research of which a preliminary account was given is being carried out in the physical laboratories of the Owens College. 'Mr. Swinburne's contribution appeared in Engineering, August and September 1903. The subjects under investigation are:-The effect of the diameter of the cordite, of the charging density, and of the shape of the enclosure. 1 The curves of rise and fall of pressure are for each explosion automatically recorded, the high-pressure recorder described at a previous meeting 1 being used for this work. Attention was drawn to the dangerous vibrations which are set up when the charge is not uniformly distributed throughout the enclosure. 3. Granular and Spicular Structure in Solids. By G. T. BEILBY. In a communication made to the British Association in 19012 I drew atten tion to certain facts which had apparently escaped the notice of other observers in micro-metallurgy. It was shown that transparence in metals is not only found in such specially attenuated forms as thin leaves or films deposited on glass, but that it is an intrinsic property of the metal even in its more massive forms. It was further shown that metal surfaces under obliquely reflected light exhibit a remarkably uniform granular or spicular structure which appears to be quite distinct from the crystalline structure revealed by the etching methods of micrometallurgy. During the past two years my study of this subject has been continued and extended, and some of the results have been already published.3 The object of the present communication is to place on record such confirmation and modification of the original observations and statements as have resulted from the further study of the subject. The original statements depended on microscopic observations made by obliquely reflected light with objectives of moderate numerical aperture. This form of illumination can only be conveniently applied with objectives whose working distance is not less than 5 mm., and whose front lens is not very large. It was therefore desirable that the observations by obliquely reflected light should be supplemented by others in which different methods of illumination could be employed. A study was made of films of metal which were thin enough to transmit light freely. By transmitted light, if such films are not too thin, they show a distinct granular texture, as if the substance had been partly gathered up into minute mounds. By alternately illuminating one of these films by transmitted and by obliquely reflected light it is seen that the structure which is granular by one light is spicular by the other; the spicular appearance, therefore, is caused by a granular texture. The slightness of this texture is shown by the fact that it is visible in oblique light in metal films which are less than 10 μu in thickness. By a parallel study of the surface-layer in metals in their more massive forms it was found that this layer is in many respects distinct from the mass which it covers, being in its structure and properties similar to the thin films deposited on glass. The character of the material on which the film is supported has a considerable influence on the appearance by obliquely reflected light. In the case of massive metal the opaque highly reflecting under-surface adds a light and colour to the spicular appearance which is absent in that of the thin glass-supported films. But if due allowance is made for this the correspondence between the appearance of the two, the surface layer and the thin film, is so exact as to leave no doubt as to the identity of the structure which causes this appearance. The transparence of thin films of metal was studied by Faraday, and some of his conclusions have been confirmed by subsequent observers. His very remarkable observations on the effect of heat annealing on thin films appear to have dropped out of sight. The subject has been studied by me with the help of lenses of a resolving power much greater than any which could be obtained in Faraday's See Report, British Association, Glasgow, 1901, p. 768, and Phil. Mag. vol. iii. p. 461, 1902. * Report, 1901, P. 604. Proc. Roy. Soc. vol. lxxii. No. 481. time. The results of these recent observations confirm and extend Faraday's conclusions, and it is believed that they also supply an answer to certain questions which he raised. The condition of the greatest opacity is found in metal films which are in a state of strain, and the condition of greatest transparence is found in films which have been relieved from strain by annealing. Contrary, therefore, to the generally accepted idea, the metal in gold leaf is in its most opaque condition. Increase of transparence in metals is accompanied by diminution of reflecting power, and vice versa. This effect can be seen most distinctly in translucent films, but it is also quite evident in the surface of the more massive forms of metal. Films of gold and of platinum 200 μu in thickness have been obtained which are translucent and optically continuous. Films less than 10 μu in thickness have also been made which appear to be equally continuous and are perfectly transparent. The thinner films are practically without metallic reflecting power, while even in the thicker films the reflection is distinctly inferior to that of gold leaf. The process of annealing has been watched on the surface of metal, and the phenomena were found to be similar in kind to those which occur in films supported on glass or mica. In surface films also the increase of transparence was well marked, and the return of the metal to the more lustrous but less transparent condition of burnishing was evident. From the study of the phenomena observed in cutting, polishing, burnishing, and annealing I have been led to the conclusion that the disturbance caused by these operations temporarily confers a degree of freedom upon the molecules of the surface layer which enables them to act like a viscous fluid subject to the influence of the molecular forces as they manifest themselves in surface tension. The dimensions and the forms of the grooves, ridges, and granules on the surface give a general indication that the layer affected by this freedom and by the surface tension is many molecules in depth. It appears probable that the granular structure of the surface is largely a result of surface tension. A similar structure can readily be developed in a very thin film of a viscous fluid. If a little oil is spread on a slip of glass and then almost completely wiped off, so that it is barely visible to the unaided eye, a granular film is produced which gives a well-marked spicular appearance by obliquely reflected light. A film of varnish on a non-reflecting support shows the same structure. A thin film of fuchsin on glass shows a structure and a play of colour which might almost be mistaken for that of a feebly reflecting gold film. Films of oxide or sulphide on metal surfaces show the spicular appearance very brilliantly. This surface granulation appears to be almost universal, and I have never failed to produce it in any solid with which I have experimented. Granular or spicular structure seems to be closely associated with the deposition of solids from solution. It is seen in thin films of metal deposited either chemically or electrolytically. In precipitates formed in very dilute solutions there are three stages in the appearance of the solid: (1) spicules or spicular films of extreme thinness, (2) granules, and (3) crystals. Spicules may be formed singly; but they often result from the breaking up of the thin films which are formed at the surface of contact of the two reagents. Their pedetic movements lead to their agglomeration into granules which sink to the bottom of the containing vessel, where they become centres of attraction to the moving spicules and grow by their absorption. Till the granule has reached a certain size and mass it shows no indication of crystalline form or structure, and its form remains, under the control of surface tension, rounded and granular. When a certain size is reached the crystallic force begins to assert itself and to overpower surface tension, and the rounded form begins to develop faces and angles till finally a welldeveloped crystal is produced. MONDAY, SEPTEMBER 14. DEPARTMENT OF MATHEMATICS. The following Papers were read :— 1. On the Differential Invariants of Surfaces and of Space. 2. On Spherical Curves. By HAROLD HILTON, M.A. If the stereographic projection of a curve on a sphere from any point is a rational algebraic curve, so is its projection from any other point. One projection is derived from another by an inversion followed by a reflexion in a straight line. In general the projection of such a spherical curve is a plane curve intersecting the line at infinity only in a multiple point at each circule. If the plane curve is of the n-th degree, the spherical curve is said to be of the n-th degree. If it has nodes and cusps, touches r great circles in two distinct points, and has great circles of curvature, then m= n2 – 28 — 3x, n = m(m − 1) −2r-31, 1 = — 2 3 - 2n(n − 2) — 68 – 8x, x = 3m (m − 2) – 6r-8t. The deficiency is (n-2)2-8-x. The foci of a spherical curve (i.e. the intersections of generating lines which touch the curve) are (m-n) in number, m-n being real, and project into the foci of the projection of the curve. Every focus of a spherical curve is a focus of its evolute. Very many properties of spherical curves may be deduced from known properties of plane curves, and vice versa; the simplest cases are the circle and the spherical quartic with zero deficiency. For instance: If three small circles are drawn passing through the cusp of a spherical quartic and touching the curve, a circle can be drawn through the focus and their other three points of intersection.' Some striking theorems can be proved for the real intersections of a real cone and a sphere. If the vertex of the cone lies outside the sphere, we can by two projections reduce the curve to the intersection of a sphere and a cylinder; if the vertex lies inside the cone, we can reduce the curve to the intersection of a sphere and a cone whose vertex is at the centre of the sphere. In either case we can deduce properties of a spherical curve of the 2p-th degree which lies on a cone by means of known properties of plane curves of the p-th degree. If the vertex of the cone is at the centre of the sphere, properties of the curve may be derived from the fact that the equation of the cone may be put into the shape 0 = a1 a2 ‚ ap + A1(.22 + y2 + ≈2) B1 B2 Bp -2 + α2 (x2 + y2 + ~2)2Y1 Y2 • • • Yk−i + • • •, ... where the a's are constants and the a's, B's, y's are real linear functions of x, y, z. The foci of the intersection of the sphere and the reciprocal cone are the 2p poles of the great circles in which the planes a1 = a, ... ap=0 intersect the sphere. Projecting on to the plane we have properties of a curve which is its own inverse with respect to a circle whose centre is real, and whose radius is real or purely imaginary. In particular many interesting properties of bicircular quartics whose four real foci are concyclic may be obtained. For example: If P is any point on a bicircular quartic whose four real foci S, S', H, H' lie on a circle, and are such that the lines SS', HH' meet inside the circle, the circles SPS', HPH' make equal angles with the tangent at P.' To many of the properties of spherical curves correspond properties of curves on a conicoid. To obtain these we project stereographically from the sphere on to a plane, project orthogonally on to another plane, and then project stereographically on to a suitable conicoid. For example: If two curves of the fourth degree on an ellipsoid both touch the generating lines through four given real coplanar points, the tangents at a point of intersection of the curves are parallel to conjugate diameters of the indicatrix at that point.' 3. The Use of Tangential Coordinates. By R. W. H. T. HUDSON. There are two reasons why it is advisable that a greater use should be made of tangential co-ordinates in elementary analytical geometry. From an educational point of view they are useful in drawing out the student's power of deduction, and exciting his interest in a way in which the long and difficult problems, with which our text-books are crowded, fail to do; and, secondly, there are many theories which find their most natural expression in these coordinates, chiefly because the absolute has a less specialised form in tangential than in point coordinates. For example, it is an easy exercise to express the equation of a circle in the form k(l2 + m2) + (Gl+ Fm + C)2=0; and then, from this, the whole projective theory follows clearly. Again, to take examples from more advanced parts of the subject, the foci of the curve p(l, m, n) = 0 are given by the roots of the equation where (1, i,-)=0, + iy. The centre of a curve of class v is best defined as the polar point of the line at infinity, and has for equation From this the property of being the centroid of the points of contact of parallel tangents follows without further analysis. Great clearness is introduced into the theory of averages in connection with areas and volumes by the exclusive use of tangential coordinates. The equation of the null-conic of an area is [[(lx +my+ n)2 dx dy = 0, which may, by proper choice of axes and use of the notation of averages, be written in the form and the ellipse of inertia is the conic conjugate to the null-conic l2x2 + m2y2 —n2 = 0. Finally, the surface of floatation is a good instance. In this case, as in other cases of approximation, it is well to take the standard equation of a plane to be z+n = lx + my so that z and n are small quantities of the second order; and then the plane equa tion of the surface takes the elegant form the integral extending over the section of floatation and V being the volume immersed. 1 Math. Gazette, No. 42, Dec. 1903. |