Interpolation

“ In the first place, if the interpolation is to be more than empirical, we must have not only the experimental results, but the laws which they obey—we must in fact go through the complete process of scientific investigation. ”

“ C in the portion XN, drawing the ordinates DQ, CP, and joining the points P, Q by a line which meets the axis in D′, also constructing the point C′ by means of the tangent at P as before, we have for the required root the new limits OD′, OC′; and proceeding in like manner with the points D′, C′, and so on as often as we please, we obtain at each step two limits approximating more and more nearly to the required root OX. The process as to the point D′, translated into analysis, is the ordinate process of interpolation. ”

“ Interpolation Tables.-—All tables of proportional parts may be regarded as interpolation tables. C. Bremiker, Tafel der Proportionalteile (Berlin, 1843) , gives proportional parts to hundredths of all numbers from 70 to 699. Schrön, Logarithms, contains an interpolation table giving the first hundred multiples of all numbers from 40 to 410. Sexagesimal tables, already described, are interpolation tables where the denominator is 60 or 600. Tables of the values of binomial theorem coefficients, which are required when second and higher orders of differences are used, are described below. ”

“ A closely equivalent but more practicable formula for interpolation by differences, as devised by Lagrange, will be found in Thomson and Tait’s Elements of Natural Philosophy, p. 115.
If no column of differences shows any tendency to become zero throughout, it is an indication that the law is of a more complicated, for instance of an exponential character, so that it requires different treatment. Dr. J. Hopkinson has suggested a method of arithmetical interpolation,‍412 which is intended to avoid much that is arbitrary in the graphical method. ”

“ In the case of a trapezette, for instance, the data are the magnitudes of certain ordinates; the problem of interpolation is to determine the values of intermediate ordinates, while that of mensuration is to determine the area of the figure of which these are the ordinates. If, as is usually the case, the ordinate throughout each strip of the trapezette can be expressed approximately as an algebraical function of the abscissa, the application of the integral calculus gives the area of the figure. ”

“ An interesting example of the interpolation of a small and independent system occurs frequently in the middle of one or other of the systems AB or BC, at the place where the space covered by the systems [Pg 56] of ridges begins to broaden out very rapidly. There are two ways in which the necessary supply of ridges makes its appearance, the one is by a series of successive embranchments (Fig. 6, 1) , the other is by the insertion of an independent system, as shown in 4, 5. ”

“ As many more interpolations may be made as shall be found necessary.
Some such method, which should perhaps be called the method of Fabritius, would, I think, in most cases probably be the best for solution of equation (7) . ”

“ If we now suppose that, when we combine the systems of the two original ensembles, the forces are so modified that the energy is no longer infinite for particles in all the space forming the diaphragm, but is diminished in a part of this space, so that it is possible for particles to pass from one system to the other, this will involve a change in the function which represents the energy of the combined systems, and the equation will no longer hold. ”

“ Let us suppose that we have also a second ensemble composed of systems in which the forces are linear functions of the coördinates, and distributed in phase according to an index which is a linear function of the partial energies relating to the normal types of vibration, say Since the two ensembles are both in statistical equilibrium, the ensemble formed by combining each system of the first with each system of the second will also be in statistical equilibrium. ”

“ If we make the limits of integration and , we get the average value of in an ensemble in which the whole system is microcanonically distributed in phase, viz., where and are connected by the equation and , if given as function of , or of and , becomes in virtue of the same equation a function of alone. ”

“ ENTROPY
We have seen above that the inevitable growth in the number of complexions is the mark of irreversibility; the number of complexions at any stage can also in a certain sense be regarded as the measure, index or determinant of that stage or state of the system of elements under consideration. Any function of the number of complexions can be regarded as such measure, index or determinant. ”

“ The procedure will then consist in assuming a definite excess of energy, of entropy, and of the masses of the various components, each per unit surface, at the interface, the potential of each component being of necessity, in equilibrium, the same as it is in the adjacent masses. The interfacial transition layer thus provides in a sense a new surface-phase coexistent with those on each side of it, and having its own characteristic equation. ”

“ Now if we add in each system of this third ensemble infinitesimal conservative forces of attraction or repulsion between particles in different shells, determined by the same law for all the systems, the functions , , and will remain constants of motion, and a function differing infinitely little from will be a constant of motion. It would therefore require only an infinitesimal change in the distribution in phase of the ensemble of compound systems to make it a case of statistical equilibrium. ”

“ The essential difference of the two kinds of influence consists in this, that the bodies to which the external coördinates relate are not distributed in phase, while in the case of interaction of the systems of two ensembles, we have to regard the fact that both are distributed in phase. To find the effect produced on the ensemble with which we are principally concerned, we have therefore to consider single values of what we have called external coördinates, but an infinity of values of the internal coördinates of any other ensemble with which there is interaction. ”

“ And in general, with the restriction mentioned, we may substitute for in an -fold integral, reducing it to a simple integral, when the limits are expressed by the potential energy, and the other factor under the integral sign is a function of the potential energy, either alone or with quantities which are constant in the integration. ”

“ The interest of the ensemble which has been described lies in the fact that it may be in statistical equilbrium, both in respect to exchange of energy and exchange of particles, with other grand ensembles canonically distributed and having the same values of and of the coefficients , , etc., when the circumstances are such that exchange of energy and of particles are possible, and when equilibrium would not subsist, were it not for equal values of these constants in the two ensembles. ”

“ This precision has not been used for the sake of the mechanical part of the problem, which does not require the surface to be defined with greater nicety than we can employ in our observations, but in order to give determinate values to the superficial densities of energy, entropy, and the component substances, which quantities, as has been seen, play an important part in the relations between the tension of a surface of discontinuity, and the composition of the masses which it separates. ”

“ Yet among the various phases of the kinds of matter concerned, represented by the different sets of values of the variables which satisfy the fundamental equation, there is a certain class which merits especial attention. These are the phases for which the entropy has a maximum value for the same matter, as determined by the ultimate analysis of the body, with the same energy and volume. To fix our ideas let us call the proximate components and the ultimate components and let denote the quantities of the former, and the quantities of the latter. ”

“ The differential equation relating to average values in the ensemble is found to correspond with the fundamental equation of thermodynamics; the modulus of distribution of ensembles turns out to be analogous to the temperature, while the average index of probability in phase is the analogue of the entropy with reversed sign, and being a minus quantity, is found to decrease just as entropy increases. ”

“ The ratio of any element of area in the volume-pressure or the entropy-temperature diagram, or in any other in which the scale of work and heat is unity, to the corresponding element in the volume-entropy diagram is represented by or The cases in which this ratio is 0, or changes its sign, demand especial attention, as in such cases the diagrams of constant scale fail to give a satisfactory representation of the properties of the body, while no difficulty or inconvenience arises in the use of the volume-entropy diagram. ”

“ But if we imagine any material system whatever, and suppose the position of a point to be determined for every possible state of the system, by making the co-ordinates of the point equal to the total volume, entropy, and energy of the system, the points thus determined will evidently form a solid figure bounded in certain directions by the surface representing the states of dissipated energy. ”

“ Thus, if we have a canonical ensemble consisting of material points in vertical circles, the theorem of the return of systems may be applied to a part of the ensemble defined as contained in a given part of the circle. But it may not be applied in all cases to a part of the ensemble defined as contained in a given part of the circle and having a given energy. It would, in fact, express the exact opposite of the truth when the given energy is the critical value mentioned above. ”

“ A line or curve formed by the intersection of two surfaces will represent solutions in equilibrium with two solid phases (viz. those belonging to the intersecting surfaces) , and will show the conditions for the existence of univariant systems. Lastly, points formed by the intersection of three surfaces will represent invariant systems, in which a solution can exist in equilibrium with three solid phases (viz. those belonging to the three surfaces) . ”

“ It is a common error which young physicists are apt to fall into to obtain a law, a curve or a mathematical expression for given experimental limits and then to apply it to points outside those limits. This is sometimes called extrapolation. Such a process, unless carefully guarded, ceases to be a reasoning process and becomes one of pure imagination specially liable to error when the distance is too great. ”

“ Let us consider an ensemble in which the number of systems having particles of the several kinds, and having values of their coördinates and momenta lying between the limits and , and , etc., is represented by the expression
where , , , are constants, denoting the total number of systems in the ensemble. The expression evidently represents the density-in-phase of the ensemble within the limits described, that is, for a phase specifically defined. The expression is therefore the probability-coefficient for a phase specifically defined. ”

“ After successive differentiation there is obtained a differential equation of the second order and its general integral is which determines the general dependence of entropy on probability. The universal integration constant is the same for a terrestrial system as for a cosmical system, and when its numerical value is known for either system it will be known for the other ”

“ When we consider the behavior of aggregates we recognize at once a general, empirical law, which has also been called the one physical axiom, namely, that all natural processes are essentially irreversible. ”

“ It may be added that, as the volume, entropy, and energy of a body are equal to the sums of the volumes, entropies, and energies of its parts, if the surface should be constructed for bodies differing in quantity but not in kind of matter, the different surfaces thus formed would be similar to one another, their linear dimensions being proportional to the quantities of matter. ”

“ Thus we have the conclusion that by the process of inversion we get from a distribution in equilibrium, on a conductor of any form, an induced distribution on the inverse surface supposed insulated and conducting; and conversely we obtain from a given induced distribution on an insulated conducting surface, a natural equilibrium distribution on the inverse surface. In each case the inducing charge is situated at the centre of inversion. ”

“ They resemble each other more than they do any one irreversible process. This is evident from an examination of the differential equations which control them; the differential with respect to time is always of an even order, because the essential sign of time can be reversed. ”