Convexity

“ At every degree of obliquity from 0° to 90°, that is, from a vertical to a horizontal view of it, the elliptical margin of the convexity will always be visible, which is impossible in a real convexity, and the elevated apex will gradually sink till the elliptical margin becomes a straight line, and the imaginary convexity completely levelled. ”

“ Secondly, a convex dial at a consider able height from the ground looks even more convex than it really is, because the lines of sight from the middle and the top of the dial make a smaller angle with the eye than the lines from the middle and the bottom, in proportion to the degree of convexity. ”

“ Of all the operations of mechanical art, that of turning is the most perfect. If two surfaces are worked against each other, whatever may have been their figure at the commencement, there exists a tendency in them both to become portions of spheres. Either of them may become convex, and the other concave, with various degrees of curvature. A plane surface is the line of separation between convexity and concavity, and is most difficult to hit; it is more easy to make a good circle than to produce a straight line. ”

“ But man is led by his passions, and particularly by hope and fear."
When parallel rays fall upon a convex mirror, they are scattered and dispersed in all directions, and the image of an object reflected in a convex mirror appears to be very small, being reduced in size because the reflected picture i m is nearer the surface of the mirror than the object a b. No. 1. (Fig. 267.)
Fig. 267.
a b, d h. (No. 2) represent two parallel rays incident on the convex surface b h, the one (a b) perpendicularly, the other (d h) obliquely. c is the centre of convexity. ”

“ As we may make and for any state of the body which we may choose, we may place the origin of co-ordinates at any point in the plane . Again, it is evident from the form of equation (1) that whatever changes we may make in the units in which volume, entropy, and energy are measured, it will always be possible to make such changes in the units of temperature and pressure, that the equation will hold true in its present form, without the introduction of constants. It is easy to see how a change of the units of volume, entropy, and energy would affect the surface. ”

“ And after some little thinking, it occurred to me that if this medium be of the same nature as our atmosphere and other gaseous bodies, that is, if it be capable of being expanded by heat and contracted by cold, then the sun being a very hot body, and the heat being so much the greater the nearer we are to him, the density of the medium ought therefore to decrease with a decreasing and increase with an increasing distance, the same as Newton would have it. ”

“ But, these motions remaining the same, their circuits of motion are reduced, the frequency of contact is increased, and heat and light are evolved from the friction. The tendency of all matter under the law of gravitation, considered as an unexplained fact, is toward concentration. The evolution of heat is rather the check put upon this tendency, and, in so far as it exerts any influence, it exerts it in a direction the reverse of gravitation. There is a perpetual and rhythmic antagonism between the forces of integration and disintegration. ”

“ The statical energy of the medium at an instant of no velocity must be equal to its kinetic energy at an instant of no displacement. Let us examine each of these quantities, and consider the equation which expresses their equality.
6. Since in every part of an element the irregular as well as the regular part of the displacement is entirely determined (for light of a given period) by the values of the statical energy of the element must be a quadratic function of say
where etc. depend only on the nature of the medium and the period of oscillation. ”

“ Suppose the entire atmosphere to be, from the bottom to the top, nearly of the same temperature, and in a perfect state of the quiescent equilibrium, and that from some cause a certain volume of air above has its specific gravity so changed that it commences to descend. As it descends the pressure upon it increases—and air, being compressed, contracts and gives out heat. A like volume ascends to take its place, and in ascending it expands and grows cool. ”

“ If our surface is plane to within a short distance of the edge, and is there overcorrected, or convex, the test shows it, as in Fig. 10. If the whole surface is regularly convex, then concentric rings of a breadth determined by the approach to a perfect plane are seen. If concave, a similar phenomenon is exhibited, except in the case of the convex, the broader rings are near the center, while in the concave they are nearer the edge. In lowering the eye while observing the plates, the rings of the convex plate will advance outward, those of the concave inward. ”

“ The word force is foreign to the subject. Yet we find that the mathematical laws of the uniform motion of heat in homogeneous media are identical in form with those of attractions varying inversely as the square of the distance. We have only to substitute source of heat for centre of attraction, flow of heat for accelerating effect of attraction at any point, and temperature for potential, and the solution of a problem in attractions is transformed into that of a problem in heat. ”

“ It has been found that if a semi-globular cup of thin metal is exposed to the wind, the pressure upon the round151 or convex side of the hemisphere is equal to two thirds of that upon the hollow or concave side. By placing four such cups upon cross-arms, and the arms on a pivot, the wind, from whatever quarter it may come, will always blow them round with their convex faces foremost; and they will move with one third of the actual velocity of the wind. ”

“ The air is an elastic fluid; and the peculiar property of elastic bodies is to resume, after compression, their original dimensions; and you must consider the air of which the atmosphere is composed as existing in a state of compression, for its particles being drawn towards the earth by gravity, are brought closer together than they would otherwise be, but the spring or elasticity of the air by which it endeavours to resist compression, gives it a constant tendency to expand itself, so as to resume the dimensions it would naturally have, if not under the influence of gravity. ”

“ It forms the surface of dissipated energy. If we consider all the points representing the volume, entropy, and energy of the body in every possible state, whether of equilibrium or not, these points will form a solid figure unbounded in some directions, but bounded in others by this surface. [10]
The lines traced upon the primitive surface by the rolling double tangent plane, which have been called the limit of absolute stability, do not end at the vertices of the triangle which represents a mixture of those states. ”

“ From the mutual attraction of the atoms of a diffused mass whose form is unsymmetrical, there results not only condensation but rotation: gravitation simultaneously generates both the centripetal and the centrifugal forces. While the condensation and the rate of rotation are progressively increasing, the approach of the atoms necessarily generates a progressively increasing temperature. As this temperature rises, light begins to be evolved; and ultimately there results a revolving sphere of fluid matter radiating intense heat and light—a sun. ”

“ And we will extend the principle further, and contend that all the phenomena of nature are consequences of the constant tendency of matter to conform to this principle of equilibrium, when suffering temporary derangement from the operation of other laws. That throughout the system of nature, equal spaces possess equal force. That what we call temperature, is nothing more than the motion of equilibrium or atomic momentum of space; or, in other words, that if all space were fluid, and in a state of equilibrium, the product of each atom of equal volume, by its motion would be a constant quality. ”

“ Let us consider a hot spherical body, whose energy we wish to communicate to another equal sphere, absolutely cold, and suppose for a moment that they have a thin outside crust absolutely impermeable to heat, and that the mass of each sphere is concentrated in a single little ball, to whose motion the heat is due, the ball in the cold sphere will of course lie still at the bottom, while the other is flying about at an inconceivable rate ”

“ It seems probable that while the diffusion is great and the force of mutual gravitation small, the chief source of heat is combination of units of matter, simpler than any known to us, into such units of matter as those we know; while, conversely, when there has been reached close aggregation, the chief source of heat is gravitation, with consequent pressure and gradual contraction. ”

“ Each of these sensible motions, produced directly or indirectly by integration of those insensible motions communicated from the Sun, becomes, as we have seen, divided and subdivided into motions less and less sensible; until it is finally reduced to insensible motions, and radiated from the Earth in the shape of thermal undulations. In their totality, these complex 455movements of aerial, liquid, and solid matter on the Earth’s crust, constitute a dependent moving equilibrium. As we before saw, there is traceable throughout them an involved combination of rhythms. ”

“ Let us first apply the elastic theory to the case of the so-called vacuum. If we write for the amplitude measured in the middle between two nodal planes, the velocities of displacement will be as and the kinetic energy will be represented by where is a constant depending on the density of the medium. The potential energy, which consists in distortion of the medium, may be represented by where is a constant depending on the rigidity of the medium. ”

“ We may hence fairly conclude that there might be a fluid medium pervading the heavens, and all bodies, of such activity that no sensible difference could be observed in the intensity of its action on bodies in a state of quiescence, or moving with a velocity not only six million, but several million million times greater than that of light ! ”

“ Let us now seek the complete value of the variation of the energy of a system of heterogeneous fluid masses, in which the influence of gravity and of the surfaces of discontinuity shall be included, and deduce from it the conditions of internal equilibrium for such a system. ”

“ In like manner, the solution of the equations subsisting between falling force and motion on the one hand and heat on the other requires an answer to the question, How great is the quantity of heat which corresponds to a given quantity of motion or falling force? For instance, we must ascertain how high a given weight requires to be raised above the ground in order that its falling force maybe equivalent to the raising of the temperature of an equal weight of water from 0 degrees to 1 degrees centigrade. ”

“ For instance, I believe that the mass of the earth generally, and therefore, also, in an especial way, its more considerable continents, do not exercise a constant and invariable force of attraction, but that this attractive force manifests itself by a certain pulse, which, according to intrinsic, necessary, and probably, also, accidental external causes, increases or decreases. Though all attempts by other objects to determine this oscillation may be too limited and rude, the atmosphere furnishes a standard both delicate and large enough to test their silent operations. ”

“ In the discussion of the transfer of heat by conduction it is assumed as a principle, borne out by experiment, that the flow of heat between two very near parallel planes, drawn in a substance, is proportional to the difference of temperature between those planes, or that the flow of heat across a plane is proportional to the rate of fall of temperature across that plane. ”

“ Now, the increase of the square of velocity corresponds, in our theory, to a rise of temperature, and in this way we can explain the effect of warming the gas, and also the law discovered by Charles, that the proportional expansion of all gases between given temperatures is the same.
The dynamical theory also tells us what will happen if molecules of different masses are allowed to knock about together. The greater masses will go slower than the smaller ones, so that, on an average, every molecule, great or small, will have the same energy of motion. ”

“ Or, if a series of isodynamics be drawn for equal infinitesimal differences of energy, any series of horizontal lines will be divided into segments inversely proportional to the pressure, and any series of vertical lines into segments inversely proportional to the temperature. We see by equations (13) and (14) , that for a motion parallel to the axis of volume, the heat received is 0 and the work done is equal to the decrease of energy, while for a motion parallel to the axis of entropy, the work done is 0, and the heat received is equal to the increase of the energy. ”

“ It may be defined as being in convective equilibrium while in such a condition. Now, if a body of air at the top be cooled a few degrees, it contracts and becomes relatively heavier. Because it is heavier, it begins to drop. ”

“ To substitute real examples for symbolical ones, the following are three uniformities, or call them laws of nature: the law that air has weight, the law that pressure on a fluid is propagated equally in all directions, and the law that pressure in one direction, not opposed by equal pressure in the contrary direction, produces motion, which does not cease until equilibrium is restored. From these three uniformities we should be able to predict another uniformity, namely, the rise of the mercury in the Torricellian tube. ”