can never be followed into all their ultimate results. 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 neutralised 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 neutralise 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 perfect mathematical analysis, absolute darkness may be shown to result from the interference of the rays. In this case the natural laws concerned, those of reflection and refraction, are accurately known, and the only difficulty consists in developing their full 1 consequences. 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 properties of matter are exhibited in the simplest 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 gases nearly obey the law of Boyle and Mariotte; 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 con 1 Lloyd's Lectures on the Wave Theory, pp. 82, 83. tinuity 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 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. In the more condensable 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 condensable, or examined at a temperature more removed from its liquefying point. Thus the perfect gas must have an infinitely high temperature. According to Dalton's law each gas in a mixture retains its own properties unaffected by the presence of any other gas.2 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.3 Mathematical Principles of Approximation. The approximate character of physical science will be rendered more plain if we consider it from a mathematical point of view. Throughout quantitative investigations we deal with the relation of one quantity to other quantities, 1 Jamin, Cours de Physique, vol. i. pp. 283-288. 2 Joule and Thomson, Philosophical Transactions, 1854, vol. cxliv. P. 337. 3 The properties of a perfect gas have been described by Rankine, Transactions of the Royal Society of Edinburgh, vol. xxv. p. 561. of which it is a function; but the subject is 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 quantities, the values of which depend upon the successive powers of the variable quantity. If y be a function of x then we may say that y = 10. A+ Bx + C x2 + Dx3 + Ex1.... 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 coefficients A, B, C, &c., may be zero or negative; but whatever they be they are fixed. The quantity on the other hand may be made what we like, being variable. Suppose, in the first place, that x and y are both lengths. Let us assume that part of an inch is the least that we can take note of. Then when is one hundredth of an inch, we have a2 = 10.000, and if C be less than unity, the term C will be inappreciable, being less than we can measure. Unless any of the quantities D, E, &c., should happen to be very great, it is evident that all the succeeding terms will also be inappreciable, because the powers of x become rapidly smaller in geometrical ratio. Thus when a is made small enough the quantity y seems to obey the equation 1 y = A+ Bx. 1 If x should be still less, if it should become as small, for instance, as 1.000.000 of an inch, and B should not be very great, then y would appear to be the fixed quantity A, and would not seem to vary with x at all. On the other hand, were x to grow greater, say equal to inch, and C not be very small, the term Ca2 would become appreciable, and the law would now be more complicated. We can invert the mode of viewing this question, and suppose that while the quantity y undergoes variations depending on many powers of x, our power of detecting the changes of value is more or less acute. While our powers of observation remain very rude we may be unable to detect any change in the quantity at all, that is to say, Bx may always be too small to come within our notice, just as in former days the fixed stars were so called because they remained at apparently fixed distances from each other. With the use of telescopes and micrometers we become able to detect the existence of some motion, so that the distance of one star from another may be expressed by A+ Bx, the term including a being still inappreciable. Under these circumstances the star will seem to move uniformly, or in simple proportion to the time x. With much improved means of measurement it will probably be found that this uniformity of motion is only apparent, and that there exists some acceleration or retardation. More careful investigation will show the law to be more and more complicated than was previously supposed. There is yet another way of explaining the apparent results of a complicated law. If we take any curve and regard a portion of it free from any kind of discontinuity, we may represent the character of such portion by an equation of the form y A+ Bx + С x2 + D x3 + Restrict the attention to a very small portion of the curve, and the eye will be unable to distinguish its difference from a straight line, which amounts to saying that in the portion examined the term Ca2 has no value appreciable by the eye. Take a larger portion of the curve and it will be apparent that it possesses curvature, but it will be possible to draw a parabola or ellipse so that the curve shall apparently coincide with a portion of that parabola or ellipse. In the same way if we take larger and larger arcs of the curve it will assume the character successively of a curve of the third, fourth, and perhaps higher degrees; that is to say, it corresponds to equations involving the third, fourth, and higher powers of the variable quantity. We have arrived then at the conclusion that every phenomenon, when its amount can only be rudely measured, will either be of fixed amount, or will seem to vary uniformly like the distance between two inclined straight lines. More exact measurement may show the error of this first assumption, and the variation will then appear to be like that of the distance between a straight line and a parabola or ellipse. We may afterwards find that a curve of the third or higher degrees is really required to represent the variation. I propose to call the variation of a quantity linear, elliptic, cubic, quartic, quintic, &c., according as it is discovered to involve the first, second, third, fourth, fifth, or higher powers of the variable. It is a general rule in quantitative investigation that we commence by discovering linear, and afterwards proceed to elliptic or more complicated laws of variation. The approximate curves which we employ are all, according to De Morgan's use of the name, parabolas of some order or other; and since the common parabola of the second order is approximately the same as a very elongated ellipse, and is in fact an infinitely elongated ellipse, it is convenient and proper to call variation of the second order elliptic. It might also be called quadric variation. As regards many important phenomena we are yet only in the first stage of approximation. We know that the sun and many so-called fixed stars, especially 61 Cygni, have a proper motion through space, and the direction of this motion at the present time is known with some degree of accuracy. But it is hardly consistent with the theory of gravity that the path of any body should really be a straight line. Hence, we must regard a rectilinear path as only a provisional description of the motion, and look forward to the time when its curvature will be detected, though centuries perhaps must first elapse. We are accustomed to assume that on the surface of the earth the force of gravity is uniform, because the variation is of so slight an amount that we are scarcely able to detect it. But supposing we could measure the variation, we should find it simply proportional to the height. Taking the earth's radius to be unity, let h be the height at which we measure the force of gravity. Then by the well-known law of the inverse square, that force will be proportional to g or to g (12h+3h2 (1 + h)2' But at all heights to which we can attain h will be so small a fraction of the earth's radius that 3 h2 will be inappreciable, and the force of gravity will seem to follow the law of linear variation, being proportional 2 h. |