Elastic band

“ If the piston, which in the experiment described has been working vertically, be made to work horizontally, a series of essentially similar results are obtained. When the piston is worked horizontally, the anterior and posterior elastic bands require to be of nearly the same strength, whereas the inferior elastic band requires to be much stronger than the superior one, to counteract the very decided tendency the wing has to fly upwards. ”

“ The frequencies of the series of very close lines which constitute a single band in a banded spectrum are connected by a law of quite different type, namely, in the simpler cases n2= A−Bm2. It may be remarked that this is the kind of relation that would apply to a row of independent similar vibrators in which the neighbours exert slight mutual influence of elastic type. If ξ denote displacement and x distance along the row, the equation d 2ξ/dt2+k2ξ=−gd 2ξ/dx2 would represent the general features of their vibration, the right-hand side arising from the mutual elastic influences. ”

“ At the lower front part the band, which is there soft and fleshy, or rather like soft thick skin, is about five inches long, and would be elastic, were it not for a thick rope-like cartilaginous or gristly substance, which forms the upper part of the band, and which is not above three inches long. The band is probably two inches thick at the upper part, and above an inch at the lower part. The back part of the band, which is rounded from a thickening at the places where it grows from each body, is not so long as the front part, which is comparatively flat. ”

“ If so, the agreement between the elasticity of the medium as calculated from the rapid alternations of luminous vibrations, and as found by the slow process of electrical experiments, shows how perfect and regular the elastic properties of the medium must be when not encumbered with any matter denser than air. If the same character of the elasticity is retained in dense transparent bodies, it appears that the square of the index of refraction is equal to the product of the specific dielectric capacity and the specific magnetic capacity. ”

“ It has already been explained that a true wave can only be created in a medium which elastically resists some kind of deformation, and persists in motion in virtue of inertia. When any material possesses such a quality of resistance to some kind of strain or deformation of such a character that the deformation disappears when the force creating it is withdrawn, it is called an elastic material. ”

“ Assuming a medium to be made up of spherical, or nearly spherical, cells, and that, when magnetic force is transmitted, these cells are made to rotate about diameters coinciding in direction with the lines of force, the tension along those lines, and the pressure at right angles to them, are accounted for by the tendency of a rotating elastic sphere [303] to contract along its polar axis and expand equatorially so as to form an oblate spheroid. ”

“ The yield-point may be raised, as, for instance, in wire which has been hardened by stretching, but the elasticity is much impaired, and it is only within very narrow limits, if at all, that proportionality between stress and strain is found. Subsequent prolonged rest gradually restores the elasticity, and after a sufficient number of weeks or months the metal is found to be elastic up to a point which may be much higher than the original elastic limit. ”

“ The elasticity of all the diameters of any plane perpendicular to the axis of a prism of rock crystal, may be considered as being sensibly the same.
2nd. All the planes parallel to the axis are far from possessing the same elastic state; but if any three of these planes be taken, restricting ourselves only to this condition, that the angles which they form with each other are equal, then their elastic state is the same. ”

“ Tests of elasticity are made by subjecting wire of different materials but of the same dimensions to the same tension. The one changing least is said to have the greatest elastic force or elasticity. If greater forces are applied to the wire and then removed, one will finally be found that will permanently stretch the wire so that it will not return exactly to the former length. ”

“ A medium, in order to be available at all, must transmit actions across it in time ; therefore there must be postulated for it the property of inertia, — of the same kind as ordinary matter possesses, for there can hardly be a more general kind, — and also the property of elasticity or statical resistance to change either of position or of form. In ordinary matter the elasticity has reference solely to deformation; while in the constitution here assumed for the æther there is perfect fluidity as regards form, but elastic resistance to rotational displacement. ”

“ In these illustrations and models we have different portions of ideal rigid matter acting upon one another, by normal pressure at mathematical points of contact—of course no forces of friction are supposed. It is exceedingly interesting to see how thus, with no other postulates than inertia, rigidity, and mutual impenetrability, we can thoroughly model not only an elastic solid, and any combination of elastic solids, but so complex and recondite a phenomenon as the passage of polarized light through a magnetic field. ”

“ It may be noted that in an iron or steel rod the wave-velocity given by (9) amounts roughly to about five kilometres per second.
The theory of plane elastic waves in an unlimited medium, whether fluid or solid, leads to differential equations of exactly the same type. Thus in the case of a fluid medium, if the displacement normal to the wave-fronts be a function of t and x, only, the equation of motion of a thin stratum initially bounded by the planes x and x+δx is
where p is the pressure, and ρ0 the undisturbed density. ”

“ The limits of perfect elasticity as regards change of shape, on the other hand, are very low, if they exist at all, for glasses and other hard, brittle solids; but a class of metals including copper, brass, steel, platinum are very perfectly elastic as regards distortion, provided that the distortion is not too great. The question can be tested by observation of the torsional elasticity of thin fibres or wires. ”

“ Iron may be connected by similar joints as for copper, or by screw joints as for gas pipes. No white lead to be used, it being a bad conductor.
24. Iron flat bands may be connected by rivets or screws, working in slots, to provide for expansion, each surface in contact being at least six times the sectional area of band.
25. Copper bands to be similarly connected. Joints between different metals may be soldered, screwed, or rivetted, the extent of surface in contact being regulated by the dimensions of the metal of the least conducting power. ”

“ In a simple train of waves in a transparent medium, the potential energy, on the elastic theory, may be divided into two parts, of which one is due to that general deformation of the ether which is represented by the equations of wave-motion, and the other to those deformations which are caused by the interference of the ponderable particles with the wave-motion, and to such displacements of the ponderable matter as may be caused, in some cases at least, by the motion of the ether. ”

“ It possesses so-called elasticity of bulk. In the case of water having a free surface there is, as we have seen, a resistance to any change of level in the surface. This may be called an elasticity of surface form. Whenever an elastic material is strained or deformed, energy has to be expended on it to create the deformation. Thus to wind up a watch-spring, stretch a piece of indiarubber, compress some air, or bend a bow, requires an energy expenditure. ”

“ What is the elastic curve?—the curve into which an elastic line forms itself under the pressure of a weight—is a problem which had been proposed by Galileo, and was fully solved, as a mathematical problem, by Euler and others.
But beams in practice are not mere lines: they are solids. And their resistance to flexure, and the amount of it, depends upon the resistance of their internal parts to extension and compression, and is different for different substances. ”

“ For the very narrow elastic fabrics, which require considerable roller pressure to hold the web snug and firm while weaving, and where it is necessary to make very accurate joinings after a break has occurred, a better movement is one in which the web roll is placed on the main take-up shaft in the form of a sleeve. ”

“ The elastic state of these plates cannot present such remarkable differences as those we have observed in the preceding series; for, being all cut round the axis of greatest elasticity, they can only contain in their plane that of least or that of mean elasticity, or lastly, those intermediate between these limits, which do not vary greatly from each other. ”

“ Elastic Solid Theory.—In the earlier days of the wave theory it was supposed that the whole of space was filled with something which acted like an elastic solid material in which the vibrations of the atoms of a luminous body started waves in all directions, just as the vibrations of a marble embedded in a jelly would send out waves through the jelly. ”

“ The case is quite different in the elastic theory. Here the force from outside necessary to maintain in any element of volume the displacement is represented by In transparent bodies, therefore, it is directed toward the center. In metals, there is a component in the direction of the motion while the radial part of the force changes its direction and is often many times greater than the opposite force in transparent bodies. ”

“ In making the flat braid, when the individual rubber threads pass through the separate holes on the outer part of the bed plate of the machine, they all gather to one common center after they are through. This causes the rubber threads, when they are at a very high tension, to draw at a very acute angle at the edge of the hole, which necessitates running them at a lower tension than is desirable for economy. ”

“ In braiding, the threads move both across and lengthwise of the fabric, taking the place of both warp and filling, therefore increased contraction, no matter whether it is produced from the use of heavier rubber, or changed material or take-up, results in greater width, inasmuch as the lengthwise position of the threads is brought into a new position which is more of a crosswise formation. ”

“ Finite Bending of Thin Rod.—The equation
curvature = bending moment ÷ flexural rigidity may also be applied to the problem of the flexure in a principal plane of a very thin rod or wire, for which the curvature need not be small. When the forces that produce the flexure are applied at the ends only, the curve into which the central-line is bent is one of a definite family of curves, to which the name elastica has been given, and there is a division of the family into two species according as the external forces are applied directly to the ends or are applied to rigid arms attached to the ends ”

“ Now in an elastic medium permeated by centres of permanent intrinsic strain, whether it be the rotational æther with its contained electrons, or an ordinary elastic solid permeated by polar strain-nuclei as described above, the specification of the strain-energy of the medium involves a mathematical function, not only of the displacement at each material point of the medium, but also of the positions of these intrinsic strain-centres which can move independently through it. ”

“ The central portions of the band carrying the load, being subjected to great wear and tear, are often made of solid india-rubber extending to nearly half the thickness of the band in the middle, and tapering off towards the edges, while the surface facing the guide rollers is of insertion coated with india-rubber. Bands properly prepared and stretched will bear a strain of 3 tons to the square inch. ”

“ If db represents twice the width of the slit, (4) gives a measure of the resulting confusion in the bands. The important point is that the slit must be made narrower as n increases if the bands are to retain the same degree of distinctness.
§ 6. Achromatic Interference Bands.—We have already seen that in the ordinary arrangement, where the source is of white light entering through a narrow slit, the heterogeneity of the light forbids the visibility of more than a few bands. The scale 688 of the various band-systems is proportional to λ. ”

“ If any one of these strands or springs is chafed and weakened, it lessens the contracting power, and the result is that the weakened or less contracted part is of relatively greater length than the parts where the rubber threads have retained their full power. ”

“ They then reappeared and became narrower with increasing flexure. This compensation increased the sensibility so that a moderate pinch of the prism by the fingers gave a marked shift of the central band which could be observed by means of the two cross-wires already referred to. However, the compensation was not sufficiently satisfactory to obtain good matches with the half-shade. Furthermore, the slightest flexure of the trough or deviation of the beam of sunlight caused the "match" to change in the one case or the band to shift in the other, owing to the narrowness of the beam. ”

“ To reduce the friction of the strings which operate in the several outside compart boards, where the strain and wear are particularly acute, and also to prevent the strings from cutting into the boards themselves, it is good practice to fix strips of ground glass between the different rows of strings, just above the compart boards. ”

“ This box is nearly exhausted of air, and its elastic face supports the pressure of the atmosphere, and yields to it with elastic resistance in proportion to the amount of pressure. Thus, if the atmospheric pressure increases, the face is pressed inward; if atmospheric pressure diminishes, the elastic reaction of the metal moves the face outward. These movements are communicated to an index by suitable and very delicate mechanism, and registered in largely magnified dimensions, by the movements of this index upon the face of the dial. ”