Residual phenomena. Unexpected agreement dance of results of different trials. a very few instances, and to say nothing of the discovery of electricity and magnetism by the ancients, the peculiar smell observed in a room in which an electrical machine is kept in action, was long ago observed, but called the "smell of electricity," and thus left unexplained. The sagacity of Schönbein led to the discovery that this is due to the formation of Ozone, a most extraordinary body, of great chemical activity; whose nature is still uncertain, though the attention of chemists has for years been directed to it. 379. Slight anomalies in the motion of Uranus led Adams and Le Verrier to the discovery of a new planet; and the fact that the oscillations of a magnetized needle about its position of equilibrium are "damped" by placing a plate of copper below it, led Arago to his beautiful experiment showing a resistance to relative motion between a magnet and a piece of copper; which was first supposed to be due to magnetism in motion, but which soon received its correct explanation from Faraday, and has since been immensely extended, and applied to most important purposes. In fact, from this accidental remark about the oscillation of a needle was evolved the grand discovery of the Induction of Electrical Currents by magnets or by other currents. We need not enlarge upon this point, as in the following pages the proofs of the truth and usefulness of the principle will continually recur. Our object has been not so much to give applications as principles, and to show how to attack a new combination, with the view of separating and studying in detail the various causes which generally conspire to produce observed phenomena, even those which are apparently the simplest. 380. If on repetition several times, an experiment conor discor- tinually gives different results, it must either have been very carelessly performed, or there must be some disturbing cause not taken account of. And, on the other hand, in cases where no very great coincidence is likely on repeated trials, an unexpected degree of agreement between the results of various trials should be regarded with the utmost suspicion, as probably due to some unnoticed peculiarity of the apparatus employed. In agreement either of these cases, however, careful observation cannot fail Unexpected to detect the cause of the discrepancies or of the unexpected or discor agreement, and may possibly lead to discoveries in a totally results of unthought-of quarter. Instances of this kind may be given trials. without limit; one or two must suffice. 381. Thus, with a very good achromatic telescope a star appears to have a sensible disc. But, as it is observed that the discs of all stars appear to be of equal angular diameter, we of course suspect some common error. Limiting the aperture of the object-glass increases the appearance in question, which, on full investigation, is found to have nothing to do with discs at all. It is, in fact, a diffraction phenomenon, and will be explained in our chapters on Light. 382. Again, in measuring the velocity of Sound by experiments conducted at night with cannon, the results at one station were never found to agree exactly with those at the other; sometimes, indeed, the differences were very considerable. But a little consideration led to the remark, that on those nights in which the discordance was greatest a strong wind was blowing nearly from one station to the other. Allowing for the obvious effect of this, or rather eliminating it altogether, the mean velocities on different evenings were found to agree very closely. dance of different 383. It may perhaps be advisable to say a few words here Hypotheses. about the use of hypotheses, and especially those of very different gradations of value which are promulgated in the form of Mathematical Theories of different branches of Natural Philosophy. 384. Where, as in the case of the planetary motions and disturbances, the forces concerned are thoroughly known, the mathematical theory is absolutely true, and requires only analysis to work out its remotest details. It is thus, in general, far ahead of observation, and is competent to predict effects not yet even observed as, for instance, Lunar Inequalities due to the action of Venus upon the Earth, etc. etc., to which no amount of observation, unaided by theory, could ever have enabled us to assign the true cause. It may also, in such subjects as Geometrical Optics, be carried to developments far beyond the reach Hypotheses. of experiment; but in this science the assumed bases of the theory are only approximate; and it fails to explain in all their peculiarities even such comparatively simple phenomena as Halos and Rainbows-though it is perfectly successful for the practical purposes of the maker of microscopes and telescopes, and has enabled really scientific instrument-makers to carry the construction of optical apparatus to a degree of perfection which merely tentative processes never could have reached. Deduction of most pro from a number of observations. 385. Another class of mathematical theories, based to some extent on experiment, is at present useful, and has even in certain cases pointed to new and important results, which experiment has subsequently verified. Such are the Dynamical Theory of Heat, the Undulatory Theory of Light, etc. etc. In the former, which is based upon the conclusion from experiment that heat is a form of energy, many formulæ are at present obscure and uninterpretable, because we do not know the mechanism of the motions or distortions of the particles of bodies. Results of the theory in which these are not involved, are of course experimentally verified. The same difficulties exist in the Theory of Light. But before this obscurity can be perfectly cleared up, we must know something of the ultimate, or molecular, constitution of the bodies, or groups of molecules, at present known to us only in the aggregate. 386. A third class is well represented by the Mathematical bable result Theories of Heat (Conduction), Electricity (Statical), and Magnetism (Permanent). Although we do not know how Heat is propagated in bodies, nor what Statical Electricity or Permanent Magnetism are the laws of their fluxes and forces are as certainly known as that of Gravitation, and can therefore like it be developed to their consequences, by the application of Mathematical Analysis. The works of Fourier*, Green†, and Poisson areremarkable instances of such development. Another good example is Ampère's Theory of Electro-dynamics, *Théorie analytique de la Chaleur. Paris, 1822. Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, Nottingham, 1828. Reprinted in Crelle's Journal. Mémoires sur le Magnétisme, Mém. de l'Acad. des Sciences, 1811. of most pro not bable result from a num servations. 387. When the most probable result is required from a Deduction number of observations of the same quantity which do exactly agree, we must appeal to the mathematical theory of ber of obprobabilities to guide us to a method of combining the results of experience, so as to eliminate from them, as far as possible, the inaccuracies of observation. Of course it is to be understood that we do not here class as inaccuracies of observation any errors which may affect alike every one of a series of observations, such as the inexact determination of a zero point, or of the essential units of time and space, the personal equation of the observer, etc. The process, whatever it may be, which is to be employed in the elimination of errors, is applicable even to these, but only when several distinct series of observations have been made, with a change of instrument, or of observer, or of both. 388. We understand as inaccuracies of observation the whole class of errors which are as likely to lie in one direction as in another in successive trials, and which we may fairly presume would, on the average of an infinite number of repetitions, exactly balance each other in excess and defect. Moreover, we consider only errors of such a kind that their probability is the less the greater they are; so that such errors as an accidental reading of a wrong number of whole degrees on a divided circle (which, by the way, can in general be "probably" corrected by comparison with other observations) are not to be included. 389. Mathematically considered, the subject is by no means an easy one, and many high authorities have asserted that the reasoning employed by Laplace, Gauss, and others, is not well founded; although the results of their analysis have been generally accepted. As an excellent treatise on the subject has recently been published by Airy, it is not necessary for us to do more than to sketch in the most cursory manner a simple and apparently satisfactory method of arriving at what is called the Method of Least Squares. 390. Supposing the zero-point and the graduation of an instrument (micrometer, mural circle, thermometer, electrometer, of most pro Deduction galvanometer, etc.) to be absolutely accurate, successive readings bable result of the value of a quantity (linear distance, altitude of a star, temperature, potential, strength of an electric current, etc.) may, and in general do, continually differ. What is most probably the true value of the observed quantity? from a num ber of ob servations. The most probable value, in all such cases, if the observations are all equally trustworthy, will evidently be the simple mean; or if they are not equally trustworthy, the mean found by attributing weights to the several observations in proportion to their presumed exactness. But if several such means have been taken, or several single observations, and if these several means or observations have been differently qualified for the determination of the sought quantity (some of them being likely to give a more exact value than others), we must assign theoretically the best practical method of combining them. 391. Inaccuracies of observation are, in general, as likely to be in excess as in defect. They are also (as before observed) more likely to be small than great; and (practically) large errors are not to be expected at all, as such would come under the class of avoidable mistakes. It follows that in any one of a series of observations of the same quantity the probability of an error of magnitude x must depend upon a2, and must be expressed by some function whose value diminishes very rapidly as x increases. The probability that the error lies between x and x+dx, where dx is very small, must also be proportional to dr. Hence we may assume the probability of an error of any magnitude included in the range of x to x + dx to be φ (κ) δεκ. Now the error must be included between + and -∞. Hence, as a first condition, The consideration of a very simple case gives us the means of determining the form of the function involved in the preceding expression*. Compare Boole, Trans. R. S. E., 1857. See also Tait, Trans, R.S. E., 1864. |