medium, the dispersion of mixed light by the prism and the production of the spectrum could not take place. Some of the most striking phenomena were thus in direct conflict with the theory. The great French mathematician, Cauchy, first pointed out the true explanation, namely that all previous investigators had made an arbitrary assumption for the sake of simplifying the calculations. They had assumed that the particles of the vibrating medium are so close together that the intervals are quite inconsiderable compared with the length of the wave, or in other terms infinitely small. This hypothesis happened to be approximately true in the case of air, so that no error was discovered in experiments on sound. Had it not been so, the earlier analysts would probably have failed to give any solution, and the progress of the subject might have been retarded. Cauchy was able to make a new approximation to truth under the more difficult supposition, that the particles of the vibrating medium are situated at considerable distances, and act and react upon the neighbouring particles by attractive and repulsive forces. To calculate the rate of propagation of a disturbance in such a medium is a work of excessive difficulty. The complete solution of the problem appears indeed to be beyond human power, so that we must be content, as in the case of the planetary motions, to look forward to successive approximations. All that Cauchy could do was to show that certain mathematical terms or quantities, neglected in previous theories, became of considerable amount under the new conditions of the problem, so that there will exist a relation between the length of the wave, and the velocity at which it travels. To remove, then, the difficulties in the way of the undulatory theory of light, a new approach to probable conditions was needed P.

Lloyd's 'Lectures on the Wave Theory,' pp. 22, 23.

In a similar manner Fourier's theory of the conduction and radiation of heat was based upon the hypothesis that the quantity of heat passing along any line is simply proportional to the rate of change of temperature. But it has since been shown by Forbes that the conductivity of a body diminishes as its temperature increases. All the details of Fourier's solution therefore require modification, and the results are in the meantime to be regarded as only approximately true .

We ought to distinguish between those problems which are physically and those which are merely mathematically incomplete. In the latter case the physical law is correctly seized, but the mathematician neglects, or is more often unable to follow out the law in all its results. The law of gravitation and the principles of harmonic or undulatory movement, even supposing the data to be correct, can never be followed into all their ultimate results. Dr. Young explained the production of Newton's rings by supposing that the rays reflected from the upper and lower surfaces of a thin film of a certain thickness were in opposite phases, and thus neutralized each other. It was pointed out, however, that as the light reflected from the nearer surface must be undoubtedly a little brighter than that from the further surface, the two rays ought not to neutralize each other so completely as they are observed to do. It was finally shown by Poisson that the discrepancy arose only from incomplete solution of the problem; for the light which has once got into the film must be to a certain extent reflected backwards and forwards ad infinitum; and if we follow out this course of the light by a perfect mathematical analysis, absolute darkness may be shown to result from the interference of the rays". In such a case as this we used no physical laws 4 Tait's Thermodynamics,' p. 10.

r

Lloyd's 'Lectures on the Wave Theory,' pp. 82, 83.

but those of reflection and refraction, and the only difficulty consisted in developing their full consequences.

There is one instructive result of the theory of error which should always be borne in mind, namely that when a large variable error is combined with a small variable error, the uncertainty of the final result, as measured by its probable error, is scarcely at all affected by the small variable error. Accordingly our efforts at accuracy must be devoted to the sources of error in the order of their magnitude. There is no use in making instruments to measure the heat of the sun with the last degree of accuracy, when the varying transparency of the atmosphere produces uncertainties of far greater amount. It is needless to observe a comet or other heavenly body with the very finest instruments if it appears low down on the horizon, where the atmospheric refraction is not accurately determinate. In short, minuter variable sources of error may be entirely neglected, so long as those of a considerably greater amount remain beyond our powers of correction.

Discovery of Hypothetically Simple Laws.

In some branches of science we meet with natural laws of a simple character which are in a certain point of view exactly true and yet can never be manifested as exactly true in natural phenomena. Such, for instance, are the laws concerning what is called a perfect gas. The gaseous state of matter is that in which the general properties of matter are exhibited in the simplest and most general manner. There is much advantage accordingly in approaching the question of molecular mechanics from this side. But when we ask the question-What is a gas? the answer must be a hypothetical one. Finding that

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Airy, Philosophical Transactions,' (1856) vol. cxlvi. p. 324.

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gases nearly obey the law of Boyle and Marriotte; that they nearly expand by heat at the uniform rate of one part in 2729 of their volume at o° for each degree centigrade; and that they more nearly fulfil these conditions the more distant the point of temperature at which we examine them from the liquefying point, we pass by the principle of continuity to the conception of a perfect gas. Such a gas would probably consist of atoms of matter at so great a distance from each other as to exert no attractive forces upon each other; but for this condition to be exactly fulfilled the distances must be infinite, so that an absolutely perfect gas cannot exist. But the perfect gas is not merely a limit to which we may approach, it is a limit passed by at least one real gas. It has been shown by Despretz, Pouillet, Dulong, Arago, and finally Regnault, that all gases diverge from the Boylean law, and in nearly all cases the density of the gas increases in a somewhat greater ratio than the pressure, indicating a tendency on the part of the molecules to approximate of their own accord, and condense into liquid. In the more condensible gases such as sulphurous acid, ammonia, and cyanogen, this tendency is strongly apparent near the liquefying point. Hydrogen on the contrary diverges from the law of a perfect gas in the opposite direction, that is, the density increases less than in the ratio of the pressure. This is a singular exception, the bearing of which I am unable to comprehend.

All gases diverge again from the law of uniform expansion by heat, but the divergence is less as the gas in question is less condensible, or examined at a temperature more removed from its liquefying point. Thus the perfect gas in this respect must have an infinitely high temperature. According to Dalton's law each gas in a mixture retains its own properties wholly unaffected by the presence t Jamin, Cours de Physique,' vol. i. pp. 283-288.

of any other gas ".

This law is probably true only by approximation, but it is obvious that it would be true of the perfect gas with infinitely distant particles *.

Mathematical Principles of Approximation.

The whole subject of the approximate character of physical science will be rendered more plain if we consider it from a general mathematical point of view. Throughout quantitative investigations we deal with the relation of one quantity to certain other quantities, of which it is a function; but the subject is quite sufficiently complicated if we view one quantity as a function of one other. Now, as a general rule, a function can be developed or expressed as the sum of certain other quantities, the values of which depend upon the successive powers of the variable quantity. Thus, if y be the one quantity which is regarded as a function of x, then we may say that

У = A + B x + С x2 + D x3 + E x1 + ....

In this equation, A, B, C, D, &c., are fixed quantities, of different values in different cases. The terms may be infinite in number or after a time may cease to have any value. Any of the co-efficients A, B, C, &c., may be zero or negative; but whatever they may be they are fixed. The quantity x on the other hand may be made what we like, being variable at our will. Suppose, in the first place, that x and y are both measurable lengths. Let us assume that 10.000 part of an inch is the least that we can take note of. Then when x is one hundredth of an inch, we have x2= 10.000, and if C be less than unity, the term Ca2 will be inappreciable, being less than we u Joule and Thomson, Philosophical Transactions,' 1854, vol. cxliv. p. 337.

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The properties of a perfect gas have been described by Rankine, Transactions of the Royal Society of Edinburgh,' vol. xxv. p. 561.