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Green', and Poisson, are remarkable instances of such development. Another good example is Ampère's Theory of Electrodynamics. And this leads us to a fourth class, which, however ingenious, must be regarded as in reality pernicious rather than useful.
336. A good type of such a theory is that of Weber, which professes to supply a physical basis for Ampère's Theory of Electrodynamics, just mentioned as one of the admirable and really useful third class. Ampère contents himself with experimental data as to the action of closed currents on each other, and from these he deduces mathematically the action which an element of one current ought to exert on an element of another-if such a case could be submitted to experiment. This cannot possibly lead to confusion. But Weber goes farther, he assumes that an electric current consists in the motion of particles of two kinds of electricity moving in opposite directions through the conducting wire; and that these particles exert forces on other such particles of electricity, when in relative motion, different from those they would exert if at relative rest. In the present state of science this is wholly unwarrantable, because it is impossible to conceive that the hypothesis of two electric fluids can be true, and besides, because the conclusions are inconsistent with the Conservation of Energy, which we have numberless experimental reasons for receiving as a general principle in nature. It only adds to the danger of such theories, when they happen to explain further phenomena, as those of induced currents are explained by that of Weber. Another of this class is the Corpuscular Theory of Light, which for a time did great mischief, and which could scarcely have been justifiable unless a luminous corpuscle had been actually seen and examined. As such speculations, though dangerous, are interesting, and often beautiful (as, for instance, that of Weber), we will refer to them again under the proper heads.
337. Mathematical theories of physical forces are, in general, of one of two species. First, those in which the fundamental assumption is far more general than is necessary. Thus the celebrated equation of Laplace's Functions contains the mathematical foundation of the theories of Gravitation, Statical Electricity, Permanent Magnetism, Permanent Flux of Heat, Motion of Incompressible Fluids, etc. etc., and has therefore to be accompanied by limiting considerations when applied to any one of these subjects.
Again, there are those which are built upon a few experiments, or simple but inexact hypotheses, only; and which require to be modified in the way of extension rather than limitation. As a notable example, we may refer to the whole subject of Abstract Dynamics, which requires extensive modifications (explained in Division III.) before it can, in general, be applied to practical purposes. 338. When the most probable result is required from a number of observations of the same quantity which do not exactly agree, we must appeal to the mathematical theory of probabilities 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. But it must be explained that we do not at present 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.
Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. Nottingham, 1828. Reprinted in Crelle's Journal.
2 Mémoires sur le Magnétisme. Mém. de l'Acad. des Sciences, 1811.
339. We understand as inaccuracies of observation the whole class of errors which are as likely to lie in one direction as 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.
340. 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 sketch in the most cursory manner what is called the Method of Least Squares.
341. Supposing the zero-point and the graduation of an instrument (micrometer, mural circle, thermometer, electrometer, galvanometer, etc.) to be absolutely accurate, successive readings 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 ?
The most probable value, in all such cases, if the observations are all equally reliable, will evidently be the simple mean; or if they are not equally reliable, 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 method of combining them in practice.
342. 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 a?, and must be expressed by some function whose value diminishes very rapidly as a increases. The probability that the error lies between a and a + 8x, where do is very small, must also be proportional to dx. The law of error thus found is
h where h is a constant, indicating the degree of coarseness or delicacy
1 of the system of measurement employed. The co-efficient
V that the sum of the probabilities of all possible errors shall be unity, as it ought to be.
343. The Probable Error of an observation is a numerical quantity such that the error of the observation is as likely to exceed as to fall short of it in magnitude.
If we assume the law of error just found, and call P the probable error in one trial, we have the approximate result
P=0.477 h. 344. The probable error of any given multiple of the value of an observed quantity is evidently the same multiple of the probable error of the quantity itself.
The probable error of the sum or difference of two quantities, affected by independent errors, is the square root of the sum of the squares of their separate probable errors.
345. As above remarked, the principal use of this theory is in the deduction, from a large series of observations, of the values of the quantities sought in such a form as to be liable to the smallest probable error. As an instance—by the principles of physical astronomy, the place of a planet is calculated from assumed values of the elements of its orbit, and tabulated in the Nautical Almanac. The observed places do not exactly agree with the predicted places, for two reasons —first, the data for calculation are not exact (and in fact the main object of the observation is to correct their assumed values); second, the observation is in error to some unknown amount. Now the difference between the observed, and the calculated, places depends on the errors of assumed elements and of observation. Our methods are applied to eliminate as far as possible the second of these, and the resulting equations give the required corrections of the elements.
Thus if o be the calculated R.A. of a planet: da, de, dw, etc., the corrections required for the assumed elements: the true R.A. is
0 + Ada + Eòe + Glow + etc., where A, E, IT, etc., are approximately known. Suppose the observed R.A. to be , then
θ+ Αδα + Εδe + Πδα +
a known quantity, subject to error of observation. Every observation made gives us an equation of the same form as this, and in general the number of observations greatly exceeds that of the quantities oa, de, dw, etc., to be found.
346. The theorems of $ 344 lead to the following rule for combining any number of such equations which contain a smaller number of unknown quantities
Make the probable error of the second member the same in each equation, by the employment of a proper factor : multiply each equation by the co-efficient of x in it and add all, for one of the final equations ; and so, with reference to y, z, etc., for the others. The probable errors of the values of x, y, etc., found from these final equations will be less than those of the values derived from any other linear method of combining the equations.
This process has been called the method of Least Squares, because the values of the unknown quantities found by it are such as to render the sum of the squares of the errors of the original equations a minimum.
347. When a series of observations of the same quantity has been made at different times, or under different circumstances, the law connecting the value of the quantity with the time, or some other variable, may be derived from the results in several ways—all more or less approximate. Two of these methods, however, are so much more extensively used than the others, tha • shall devote a page or two here to a preliminary notice of them, leaving detailed instances of their application till we come to Heat, Electricity, etc.
They consist in (1) a Curve, giving a graphic representation of the relation between the ordinate and abscissa, and (2) an Empirical Formula connecting the variables.
348. Thus if the abscissae represent intervals of time, and the ordinates the corresponding height of the barometer, we may construct curves which show at a glance the dependence of barometric pressure upon the time of day; and so on. Such curves may be accurately drawn by photographic processes on a sheet of sensitive paper placed behind the mercurial column, and made to move past it with a uniform horizontal velocity by clockwork. A similar process is applied to the Temperature and Electricity of the atmosphere, and to the components of terrestrial magnetism.
349. When the observations are not, as in the last section, continuous, they give us only a series of points in the curve, from which, however, we may in general approximate very closely to the result of continuous observation by drawing, liberá manu, a curve passing through these points. This process, however, must be employed with great caution; because, unless the observations are sufficiently close to each other, most important fluctuations in the curve may escape notice. It is applicable, with abundant accuracy, to all cases where the quantity observed changes very slowly. Thus, for instance, weekly observations of the temperature at depths of from 6 to 24 feet
h(h—1) 8°F (20) +...
underground were found by Forbes sufficient for a very accurate approximation to the law of the phenomenon.
350. As an instance of the processes employed for obtaining an empirical formula, we may mention methods of Interpolation, to which the problem can always be reduced. Thus from sextant observations, at known intervals, of the altitude of the sun, it is a common problem of Astronomy to determine at what instant the altitude is greatest, and what is that greatest altitude. The first enables us to find the true solar time at the place, and the second, by the help of the Nautical Almanac, gives the latitude. The calculus of finite differences gives us formulae proper for various data; and Lagrange has shown how to obtain a very useful one by elementary algebra. In finite differences we have f(x+h)=f(«) + haf(x) +
1.2 This is useful, inasmuch as the successive differences, Af(x), Af(a), etc., are easily calculated from the tabulated results of observation, provided these have been taken for equal successive increments of x. If for values X1, 22,
Xn, a function takes the values Y1, 92, 93, Yn, Lagrange gives for it the obvious expression y
+ ... ](x - 1) (oc — 3C2)... (26—2cm). 31 (32 — xz) (21 — xz)...
(2013.) ' x - xy (cz — 3) (2C2 — )... (C2 — wn) Here it is assumed that the function required is a rational and integral one in w of the n-1th degree; and, in general, a similar limitation is in practice applied to the other formula above; for in order to find the complete expression for f(«), it is necessary to determine the values of Af(oc), AF(oc), .... If n of the co-efficients be required, so as to give the n chief terms of the general value of f(x), we must have n observed simultaneous values of x and f(x), and the expression becomes determinate and of the n– 1th degree in h.
In practice it is usually sufficient to employ at most three terms of the first series. Thus to express the length 1 of a rod of metal as depending on its temperature t, we may assume
I=1,+At-t.)+B(t-t.)?, 1, being the measured length at any temperature to. A and B are to be found by the method of least squares from values of l observed for different given values of t.
351. These formulae are practically useful for calculating the probable values of any observed element, for values of the independent variable lying within the range for which observation has given values of the element. But except for values of the independent variable either actually within this range, or not far beyond it in either direction, these formulae express functions which, in general, will differ more and more widely from the truth the further their application is pushed beyond the range of observation.
In a large class of investigations the observed element is in its