kinetic mediately to the principle of least action, and the Hamiltonian Liouville's principle with the developments by Jacobi and others; but theorem. which also establishes a very remarkable and absolutely new theorem regarding the amount of the action along any constrained course. For brevity we shall content ourselves with giving it for a single free particle, referring the reader to the original article for Liouville's complete investigation in terms of generalized co-ordinates, applicable to any conservative system whatever. Let (x, y, z) be the co-ordinates of any point through which the particle may move: V its potential energy in this position: E the sum of the potential and kinetic energies of the motion in question: A the action, from any position (x, y, z) to (x, y, z) along any course arbitrarily chosen (supposing, for instance, the particle to be guided along it by a frictionless guiding tube). Then (§ 326), the mass of the particle being taken as unity, A = fvds = √ √2 (E − V ) ↓ (dx2 + dy3 + dz3). Now let be a function of x, y, z, which satisfies the partial differential equation CHAPTER III. EXPERIENCE. Observation and experiment. Observa tion. 369. By the term Experience, in physical science, we designate, according to a suggestion of Herschel's, our means of becoming acquainted with the material universe and the laws which regulate it. In general the actions which we see ever taking place around us are complex, or due to the simultaneous action of many causes. When, as in astronomy, we endeavour to ascertain these causes by simply watching their effects, we observe; when, as in our laboratories, we interfere arbitrarily with the causes or circumstances of a phenomenon, we are said to experiment. 370. For instance, supposing that we are possessed of instrumental means of measuring time and angles, we may trace out by successive observations the relative position of the sun and earth at different instants; and (the method is not susceptible of any accuracy, but is alluded to here only for the sake of illustration) from the variations in the apparent diameter of the former we may calculate the ratios of our distances from it at those instants. We have thus a set of observations involving time, angular position with reference to the sun, and ratios of distances from it: sufficient (if numerous enough) to enable us to discover the laws which connect the variations of these co-ordinates. Similar methods may be imagined as applicable to the motion of any planet about the sun, of a satellite about its primary, or of one star about another in a binary group. 371. In general all the data of Astronomy are determined in this way, and the same may be said of such subjects as tion. Tides and Meteorology. Isothermal Lines, Lines of Equal Dip, ObservaLines of Equal Intensity, Lines of Equal "Variation” (or “Declination" as it has still less happily been sometimes called), the Connexion of Solar Spots with Terrestrial Magnetism, and a host of other data and phenomena, to be explained under the proper heads in the course of the work, are thus deducible from Observation merely. In these cases the apparatus for the gigantic experiments is found ready arranged in Nature, and all that the philosopher has to do is to watch and measure their progress to its last details. 372. Even in the instance we have chosen above, that of the planetary motions, the observed effects are complex; because, unless possibly in the case of a double star, we have no instance of the undisturbed action of one heavenly body on another; but to a first approximation the motion of a planet about the sun is found to be the same as if no other bodies than these two existed; and the approximation is sufficient to indicate the probable law of mutual action, whose full confirmation is obtained when, its truth being assumed, the disturbing effects thus calculated are allowed for, and found to account completely for the observed deviations from the consequences of the first supposition. This may serve to give an idea of the mode of obtaining the laws of phenomena, which can only be observed in a complex form-and the method can always be directly applied when one cause is known to be pre-eminent. ment. 373. Let us take cases of the other kind—in which the effects Experiare so complex that we cannot deduce the causes from the observation of combinations arranged in Nature, but must endeavour to form for ourselves other combinations which may enable us to study the effects of each cause separately, or at least with only slight modification from the interference of other causes. 374. A stone, when dropped, falls to the ground; a brick and a boulder, if dropped from the top of a cliff at the same moment, fall side by side, and reach the ground together. But a brick and a slate do not; and while the former falls in a nearly vertical direction, the latter describes a most complex Experi ment. path. A sheet of paper or a fragment of gold leaf presents even greater irregularities than the slate. But by a slight modification of the circumstances, we gain a considerable insight into the nature of the question. The paper and gold leaf, if rolled into balls, fall nearly in a vertical line. Here, then, there are evidently at least two causes at work, one which tends to make all bodies fall, and fall vertically; and another which depends on the form and substance of the body, and tends to retard its fall and alter its course from the vertical direction. How can we study the effects of the former on all bodies without sensible complication from the latter? The effects of Wind, etc., at once point out what the latter cause is, the air (whose existence we may indeed suppose to have been discovered by such effects); and to study the nature of the action of the former it is necessary to get rid of the complications arising from the presence of air. Hence the necessity for Experiment. By means of an apparatus to be afterwards described, we remove the greater part of the air from the interior of a vessel, and in that we try again our experiments on the fall of bodies; and now a general law, simple in the extreme, though most important in its consequences, is at once apparent-viz., that all bodies, of whatever size, shape, or material, if dropped side by side at the same instant, fall side by side in a space void of air. Before experiment had thus separated the phenomena, hasty philosophers had rushed to the conclusion that some bodies possess the quality of heaviness, others that of lightness, etc. Had this state of confusion remained, the law of gravitation, vigorous though its action be throughout the universe, could never have been recognised as a general principle by the human mind. Mere observation of lightning and its effects could never have led to the discovery of their relation to the phenomena presented by rubbed amber. A modification of the course of nature, such as the collecting of atmospheric electricity in our laboratories, was necessary. Without experiment we could never even have learned the existence of terrestrial magnetism. 375. When a particular agent or cause is to be studied, of expect experiments should be arranged in such a way as to lead if possible to results depending on it alone; or, if this cannot be Rules for the conduct ments. done, they should be arranged so as to show differences pro- Rules for duced by varying it. 376. Thus to determine the resistance of a wire against the conduction of electricity through it, we may measure the whole strength of current produced in it by electromotive force between its ends when the amount of this electromotive force is given, or can be ascertained. But when the wire is that of a submarine telegraph cable there is always an unknown and ever varying electromotive force between its ends, due to the earth (producing what is commonly called the "earth-current"), and to determine its resistance, the difference in the strength of the current produced by suddenly adding to or subtracting from the terrestrial electromotive force the electromotive force of a given. voltaic battery, is to be very quickly measured; and this is to be done over and over again, to eliminate the effect of variation of the earth-current during the few seconds of time which must elapse before the electrostatic induction permits the current due to the battery to reach nearly enough its full strength to practically annul error on this score. 377. Endless patience and perseverance in designing and trying different methods for investigation are necessary for the advancement of science: and indeed, in discovery, he is the most likely to succeed who, not allowing himself to be disheartened by the non-success of one form of experiment, judiciously varies his methods, and thus interrogates in every conceivably useful manner the subject of his investigations. the conduct of experi ments. phenomena. 378. A most important remark, due to Herschel, regards Residual what are called residual phenomena. When, in an experiment, all known causes being allowed for, there remain certain unexplained effects (excessively slight it may be), these must be carefully investigated, and every conceivable variation of arrangement of apparatus, etc., tried; until, if possible, we manage so to isolate the residual phenomenon as to be able to detect its cause. It is here, perhaps, that in the present state of science we may most reasonably look for extensions of our knowledge; at all events we are warranted by the recent history of Natural Philosophy in so doing. Thus, to take only |