Ρ culus of differences a correct empirical formula; for if be the first term of the series of values, and ▲p, ▲2p, 'p, &c., be the first number in each column of differences, then the mth term of the series of values will be m-I p+ msp + m2 ·A2p + m2 m-1 M-2 3 2 3 Ap+ &c. 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 cannot be correctly represented in a formula involving only a few powers of the variable. Dr. J. Hopkinson has lately suggested another method of arithmetical interpolation, which is intended to avoid, much that is arbitrary in the graphical method. His process will yield the same results in all hands, but he remarks that it has no theoretical basis to rest on. So far as we can infer the results likely to be obtained by variations beyond the limits of experiment, we must proceed upon the same principles. If possible we must detect the exact laws in action, and then trust to them as a guide when we have no experience. If not, an empirical formula of exactly the same character as those employed in interpolation is our only resource. But the reader must carefully observe that to extend our inference far beyond the limits of experience is exceedingly unsafe. Our knowledge is at the best only approximate, and takes no account of very small tendencies. Now it may, and in fact usually will, happen, that tendencies small within our limits of q 'On the Calculation of Empirical Formulæ.' 'The Messenger of Mathematics,' New Series, No. 17, 1872. observation will become perceptible or great under extreme circumstances. When the variable in our empirical formula is small, we are justified in overlooking the existence of higher powers, and taking only two or three of them. But as the variable increases, those higher powers gain in importance, and in time will yield the principal part of the value of the function. This is no mere theoretical inference. Excepting the few great primary laws of nature, such as the law of gravity, the conservation of energy, &c., there is hardly any natural law which we can trust in circumstances widely different from those with which we are practically acquainted. From the expansion or contraction, fusion or vaporisation of substances by heat at the surface of the earth, we can form a most imperfect notion of what would happen near the centre of the earth, where the pressure must almost infinitely exceed anything possible in our experiments. The physics of the earth again give us a feeble, and probably often a misleading, notion of a body like the sun, in most parts of which an almost inconceivably high temperature is united with an inconceivably high pressure. If, as is probable, there are in the realms of space many nebulæ consisting of incandescent and unoxydized vapours of metals and other elements, so highly heated perhaps that chemical composition is out of the question, we are hardly able to treat them as subjects of scientific inference. Hence arises the great importance of any experiments in which we can investigate the properties of substances under extreme circumstances of cold or heat, density or rarity, intense electric excitation, &c. It should be observed that this insecurity in extending our inferences wholly arises from the purely approximate character of our measurements. Had we the power of appreciating indefinitely small quantities, we should by the principle of continuity discover some trace of every change which a substance could undergo under unattainable circumstances. By observing, for instance, the tension of aqueous vapour between o° and 100° C., we ought theoretically to be able to infer its tension at every other temperature; but this is out of the question because we cannot really ascertain the law precisely between those temperatures. Many instances might be given to show that laws which appear to represent correctly the results of experiments within certain limits altogether fail beyond those limits. The experiments of Roscoe and Dittmar, on the absorption of gases in water" afford many interesting illustrations, especially, in the case of hydrochloric acid, the quantity of which dissolved in water under different pressures follows very closely a linear law of variation, from which however it diverges very widely at low pressures. Sir J. Herschel having deduced from various recorded observations of the double star y Virginis, an elliptic orbit for the motion of one component round the centre of gravity of both, found that for a certain time the motion of the star agreed very well with this orbit. Nevertheless a divergence began to appear by degrees, and after a time became so great that an entirely new orbit, of more than double the linear dimensions of the old one, had ultimately to be adoptedt. Illustrations of Empirical Quantitative Laws. Although our chief object in every quantitative inquiry must be to discover the exact or rational formulæ, expressing the general laws of nature applying to the subject, it is instructive to observe in how many important branches r Watts's Dictionary of Chemistry,' vol. ii. p. 790. 8 'Quarterly Journal of the Chemical Society,' vol. viii. p. 15. t 'Results of Observations at the Cape of Good Hope,' p. 293. of science, no precise laws have yet been detected. The tension of aqueous vapour at different temperatures has been determined by a succession of eminent experimentalists, Dalton, Kæmtz, Dulong, Arago, Magnus, and Regnault, and by the last mentioned the measurements were conducted with all accuracy apparently attainable at present. Yet no incontestible general law has been established. Several functions have been proposed to express the elastic force of the vapour as depending on the temperature. The first general form is that of Young, namely F = (a + b t)", in which a, b and m are unknown quantities to be determined by comparison with observation. Roche has proposed, on theoretical grounds, a complicated formula of an exponential form, and a third form of function is that of Biot, as follows-log F=a+ba'+c ß'u. I mention these formulæ particularly, because they well illustrate the feeble powers of empirical inquiry. None of the formulæ can be made to correspond exactly with experimental results, and the last two forms correspond nearly equally well. But there is very little probability that the real law has been reached, and it is highly unlikely that it will be discovered except by deduction from mechanical theory. The same remarks may be made upon any other laws except those of the most simple character. A vast amount of the most ingenious labour has been spent upon the discovery of some general law of atmospheric refraction. Tycho Brahe and Kepler commenced the inquiry: Cassini first formed a table of refractions, calculated on theoretical grounds: Newton entered into some profound investigations upon the subject: Brooke Taylor, Bouguer, Simpson, Bradley, Mayer, and Kramp successively attacked the question, which is of the highest practical importance as regards the correction of astronomical observations. Laplace next u Jamin, 'Cours de Physique,' vol. ii. p. 138. laboured on the subject without exhausting it, and Brinkley and Ivory have since treated it. A closely connected problem, that regarding the relation between the pressure and elevation in different strata of the atmosphere, has received the attention of a long succession of physicists and was most carefully investigated by Laplace. Yet no invariable and general law has been detected. The same may be said concerning the law of human mortality; abundant statistics on this subject are available, and many hypotheses more or less satisfactory have been put forward as to the general form of the curve of mortality, but it seems to be impossible to discover more than an approximate law. It may perhaps be urged that in such subjects no single invariable law can be expected. The atmosphere may be divided into several variable strata which by their unconnected changes frustrate the exact calculations of astronomers. Human life may be subject at different ages to a succession of different influences incapable of reduction under any one law. The results observed may in fact be aggregates of an immense number of separate results each governed by their own separate laws, so that the subjects may be complicated beyond the possibility of complete resolution by empirical methods. This is certainly true of the mathematical functions which must some time or other be introduced into the science of political economy. Simple Proportional Variation. When we first treat numerical results in any novel kind of investigation, our impression will probably be that one quantity varies in simple proportion to another, so as to obey the law y = mx +n. We must learn to distinguish carefully between the cases where this proportionality is really, and where it is only apparently true. When con |