= V2d. a bar at various temperatures the variation of the modulus of elasticity with temperature can be obtained. We observe N at various known temperatures of the bar, whence V is computed, and the modulus is M As l, t, and d have different values at different temperatures, the coefficient of expansion of each bar experimented on was determined, also the density of each bar at 40° C., and the dimensions and density computed for each temperature of the vibrating bar. g Though the moduli thus obtained are not so accurate as those given by other more precise methods, yet I think that the variation of the modulus with temperature is thus obtained. Tables and curves of results of the experiments on bars of glass, cast steel, Bessemer steel, brass, bell-metal, two specimens of aluminium, silver, and zinc were exhibited before the Section. The moduli of these substances are lowered by increase of temperature as follows. Heating from 0° to 100° lowers the modulus of Silver heated from 0° to 60° has its modulus lowered 2:47 per cent. of its modulus at 0°. Zinc heated from 0° to 62° has its modulus lowered 6.04 per cent. of its modulus at 0°. 9. On an Apparatus for Measuring small Strains. 10. On Mirrors of Magnetism. By Professor SILVANUS P. THOMPSON, D.Sc., F.R.S. The author has found by experiment that a sheet of iron, if sufficiently large and thick, acts magnetically as a mirror, giving virtual images of a magnet pole placed in front. The image of a pole placed at any distance in front (provided the mirror is large as compared with the distance) is situated at an equal distance behind the mirror, exactly as in optical reflexion by a plane mirror. The image of a north pole is a south pole, just as the image of a right hand is a left hand. If the magnet pole is moved away from the sheet of iron, the image moves away at an equal speed in the reverse direction. If the sheet of iron be moved up toward the magnet pole, the image moves up at double the velocity, as optical images do. If a magnet pole is placed between two parallel large sheets of iron, the effect is the same as if there were a double series of images in an indefinitely extended the images being, however, of alternate polarity. row, The action of a spherical surface of iron as a spherical mirror presents some curious points. Only virtual images are produced whether the mirror be convex or concave. There is no spherical aberration, and the images, though of reversed polarity and perverted, are not inverted. These last points are deduced from the experiments, and have not yet been independently verified. The method of experiment has been to produce magnetic poles or fields by means of coils supplied with electric currents. These poles or fields were investigated by means of an exploring coil or coils connected with a galvai Ometer, the throw of which was observed when the primary current was turned on or off. The throw obtained when the mirror was absent was compared (<) with that obtained when the mirror was present, (b) with that obtained when, the mirror being absent, the geometrical position of the coil was occupied by an actual coil of the same magnetic moment as the 'image.' Iron in thin sheet as used for tin-plate is not thick enough to form a perfect mirror. A piece of boiler-plate, inch thick and rather less than 3 feet square, formed a perfect plane mirror for poles placed not more than 4 or 5 inches away from its centre. A long solenoid 200 centimetres long, 15 centimetre diameter, uniformly wound with twelve turns per centimetre with a wire capable of carrying 15 amperes without overheating, was used in one series of experiments to produce a pole. In the case of a convex spherical surface of iron of infinite magnetic permeability, and of radius of curvature r, the object being a pole of strength m situated at a point at a distance a from the middle of the mirror surface, the image is a pole of strength behind the surface is equal to r— -m Its distance a+r дог a+r The image of an infinitely distant pole is at the centre, not as in the case of the optical image half-way between surface and centre. The following simple construction gives the position of the image of any pointpole outside. Let P be the position of the point-object, and O the centre of the spherical surface. Join PO, and with P as centre describe the arc OS. Then with centre S and OS as radius describe arc OQ cutting PO in Q.Q is the image of P. For a concave mirror the construction is reversed by finding successively the centres of the arcs. The image is a pole having strength =-m where a is a+r Both formulæ show that as r increases to infinity a = x. The whole of the experimental work has been carried out for me by Mr. Miles Walker. 11. The Volume Changes which accompany Magnetisation in Nickel Tubes. By Professor C. G. KNOTT, D.Sc. The method of experiment was similar to that employed in the determination of corresponding changes in iron and steel tubes, and already described in a former communication. Three nickel tubes, cut originally from the same solid bar, were turned and bored. They were of the same length (47 cm.) and the same external diameters (4.2 cm.). The internal diameters were as follows:-No. I., 2·543 cm.; No. II., 1.586 cm.; and No. III., 0·692 cm. The decrease of volume in the tube of widest bore (No. I.), when subjected to a longitudinal field of 600, was so large that it had to be measured with the naked eye. The liquid meniscus in the capillary tube moved outwards through a distance of 3 cm., which corresponded to a volume change of 2-4 cubic millimetres. This, with a total internal volume of 224-47 cubic centimetres, gives a dilatation of fully -10-5. In the following table some of the more striking results are indicated. Under each heading of field is a column containing the corresponding cubical dilatations for the three tubes. Thus in lowest fields the cubical dilatation is negative in Tubes I, and II. Very soon, however, it becomes positive, but changes back to negative in still higher fields, and so continues to the highest fields used. In Tube III. there is no evidence of the negative dilatation in low fields, so that with it there is only one change of sign. The thicker the wall, the higher the field in which the change of sign takes place. In moderate and high fields the changes of volume are distinctly greater in nickel than in iron or steel. 12. On Hysteresis in Iron and Steel in a Rotating Magnetic Field. By FRANCIS G. BAILY, M.A. It has long been surmised that the hysteresis in iron may have different values according to the mode of variation of the direction of magnetisation. As a deduction from Professor Ewing's molecular theory of magnetism Mr. James Swinburne pointed out that the value of the hysteresis in an alternating field should be different, and obey a different law, from that of iron in a rotating magnetic field, the latter probably being the smaller, especially at a high induction. The point has not, however, received any experimental verification until now. The experiments here described consisted in causing a powerful electromagnet to revolve on an axis concentric with the bore of the pole pieces, which formed parts of a cylinder. The magnetic field between the two pole pieces rotated with the electromagnet producing it. In the polar cavity was placed a finely laminated cylindrical armature of iron or steel carried by hollow centres in fixed supports, and held by a spring attached to the armature spindle and to the fixed support respectively. On rotating the field-magnet there is a force due to the hysteresis of the armature tending to cause it to revolve with the magnet. This is prevented by the spring, and the deflexion of the spring is a measure of the torque exerted on the armature, which is proportional to the hysteresis in the armature per reversal, and is independent of the speed of revolution of the magnet. The deflexion is ol served by the movement of a beam of light reflected from a small mirror on the 'See B.A. Reports, 1892, p. 659. armature, and on to a circular scale. Varying currents were sent round the fieldmagnet coils, producing different values of induction in the armature. At a low induction the hysteresis is small and increases but slowly, the range corresponding with the first part of the B/H curve, and the shape of the curve being similar to that given by an alternating field. With increasing induction the value of the hysteresis rises more rapidly, but does not continue to increase at a uniform rate. When B=12,000 the rate of increase of the hysteresis diminishes, though the curve still rises rapidly. When B-17,000 the curve begins to bend over, reaching a maximum at 18,000 or 18,500, and rapidly falls when the induction is further increased, until when B-21,000 the hysteresis is about one-third of its maximum value. Higher inductions could not be satisfactorily reached with the apparatus, but there was no indication of any diminution in the rate of decrease of the hysteresis. Both soft charcoal iron and hard high carbon steel were tested, and found to give the same results, the value in the steel reaching a maximum at an induction of 16,000. These experiments show that there is no simple relation between hysteresis and induction. The three stages of magnetic displacement each have a sharply defined position on the hysteresis curve, the critical value occurring just on the knee of the induction curve. The agreement of the results with the theoretical deduction constitutes a strong verification of the molecular theory of magnetism. The above results hold good for all speeds up to seventy revolutions per second, which was the highest speed obtained. 13. On the Vibrations of a Loaded Spiral Spring. The author pointed out that by comparing the two periods of vibration of a body attached to a spiral spring of small angle the ratio of the torsional and flexural rigidities of the wire or strip forming the spring could be found, and hence, if the wire were homogeneous, isotropic, and of circular section, Poisson's ratio for its material could be determined. Some experiments were shown illustrating the normal modes of vibration of such a system, and the transference of energy from up-and-down to twisting vibrations, and back again, which can be effected when the periods are nearly equal. TUESDAY, AUGUST 14. The Section was divided into three Departments. The following Papers and Report were read:- DEPARTMENT I. 1. On Fuchsian Functions. By Professor Mittag-Leffler. Professor Mittag-Leffler, after referring to a recent investigation by one of his pupils, M. G. Cassel, spoke of the advantages and inconveniences connected with the expression of automorphic functions by means of the Fuchsian or Kleinian e series. He showed how the theory of the functions may be brought into connection with that of the Abelian functions, and that an expression given by M. Schottky for a certain class of automorphic functions is, in fact, applicable to all these functions. He then referred to his own researches on the invariants of linear differential equations, and pointed out that Günther, in his paper in 'Crelle's Journal,' has not fully realised the true nature of these researches. 1894. PP 2. On Ronayne's Cubes. By Professor H. HENNESSY, F.R.S. Some years since a box containing a pair of equal cubes was placed in the author's hands, and he found that one of these could have its parts displaced so as to leave a peculiarly shaped shell, through which the second cube passed without any difficulty. Groups of twin crystals of a cubical form have been long known, but their grouping could rarely admit of such a structure as the cubes referred to present. It is manifest that a cube passed through another in the direction of the diagonal of the square would leave two triangular prisms, but in order to connect these prisms two flanges with interior sloping faces should be attached. The thickness of these flanges in two directions, as well as the angle of inclination of the sloping faces, are all connected by geometrical conditions which permit of the solution of the problem of the construction of the shell of the first cube. The author was unable to find the solution of the problem originally published by the inventor of the cubes, Mr. J. Ronayne, some time about the middle of the last century. Under these circumstances be completed the inquiry, and on comparing the results with the measured dimensions of the prisms and flanges which constitute the shell of the first cube he found the perfect concordance between the calculated and measured dimensions. When a represents the distance from a corner of the cube to the edge of the flange, and a the side of the cube, then 8, the inclination of the sloping face to the face of the cube, is found to be represented by 2a (a√2-2x) 3a2 + 4x2 - 4ax√2 sin @= The edge of each cube is 1.92 inch, and is found both by measurement and by the above formula to be 9° 45′ nearly. 3. On a Property of the Catenary. By Professor H. HENNESSY, F.R.S. In the course of inquiry into some hydraulic questions the author found that the catenary of maximum area under a given perimeter may be inscribed in a semicircle. Hence, if the radius of the semicircle is one foot, the chain hung within it when in a vertical plane will be one yard. Thus the two fundamental standards of English measure are connected with the catenary of maximum area. 4. A Complete Solution of the Problem, 'To find a Conic with respect to which two given Conics shall be Reciprocal Polars.' By J. W. Russell, M.A. In the author's 'Elementary Treatise on Pure Geometry,' p. 147, a construction is given in the case in which the given conics intersect in distinct points. This construction was extended to the cases of the conics touching or having threepoint contact. The method in the case of double contact was different. Taking U to be the common pole and AB the common chord, through U draw any line meeting the conics in PP', QQ' respectively. Let X Y be the double points of the volution PQ, P'Q' or of PQ', P'Q. Then the conic required is the conic touching UA at A and UB a B, and passing through X or Y. This method holds, when suitably modified, in the case of four-point contact. 5. The Impossibility of Trigraphic Fields of Spaces. The simplest trigraphic form is that generated by three points P, P', P" on three lines AB, A'B', A'B', these points being connected by a relation of the |