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Here is assumed that the function required is a rational and integral one in x of the n-yth 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(x), it is necessary to determine the values of Af (x), AYf (x),.... 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-ith degree in h.
In practice it is usually sufficient to employ at most three terms of the first series. Thus to express the length l of a rod of metal as depending on its temperature t, we may assume
1=1,+A(t-t.) + B(t-1.), 1 being the measured length at any temperature too 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 inde. pendent 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 nature a periodic function of the independent variable. The harmonic analysis (S 88) is suitable for all such. When the values of the independent variable for which the element has been observed are not equidifferent the co-efficients, determined according to the method of least squares, are found by a process which is necessarily very laborious; but when they are equidifferent, and especially when the difference is a submultiple of the period, the equation derived from the method of least squares becomes greatly simplified. Thus, if 8 denote an angle increasing in proportion to t, the time, through four right angles in the period, T, of the phenomenon; so that
+ B, sin 0 + B, sin 20 where A0, A,, Ap...B,, BR... are unknown co-efficients, to be determined so that flo may express the most probable value of the element, not merely at times between observations, but through all time as long as the phenomenon is strictly periodic. By taking as many of these coefficients as there are of distinct data by observation, the formula is made to agree precisely with these data. But in most applications of the method, the periodically recurring part of the phenomenon is expressible by a small number of terms of the harmonic series, and the higher terms, calculated from a great number of_data,
express either irregularities of the phenomenon not likely to recur: or errors of observation. Thus a comparatively small number of terms may give values of the element even for the very times of observation, more probable than the values actually recorded as having been observed, if the observations are numerous but not minutely accurate.
The student may exercise himself in writing out the equations to determine five, or seven, or more of the coefficients according to the method of least squares; and reducing them by proper formulae of analytical trigonometry to their simplest and most easily calculated forms where the values of 0 for which f(0) is given are equidifferent. He will thus see that when the difference is 3*, i being any integer,
, and when the number of the data is i or any multiple of it, the equations contain each of them only one of the unknown quantities : so that the method of least squares affords the most probable values of the co-efficients, by the easiest and most direct elimination.
MEASURES AND INSTRUMENTS.
352. Having seen in the preceding chapter that for the investigation of the laws of nature we must carefully watch experiments, either those gigantic ones which the universe furnishes, or others devised and executed by man for special objects—and having seen that in all such observations accurate measurements of Time, Space, Force, etc., are absolutely necessary, we may now appropriately describe a few of the more useful of the instruments employed for these pur. poses, and the various standards or units which are employed in them.
353. Before going into detail we may give a rapid résumé of the principal Standards and Instruments to be described in this chapter. As most, if not all, of them depend on physical principles to be detailed in the course of this work, we shall assume in anticipation the establishment of such principles, giving references to the future division or chapter in which the experimental demonstrations are more particularly explained. This course will entail a slight, but unavoidable, confusion-slight, because Clocks, Balances, Screws, etc., are familiar even to those who know nothing of Natural Philosophy ; unavoidable, because it is in the very nature of our subject that no one part can grow alone, each requiring for its full development the utmost resources of all the others. But if one of our departments thus borrows from others, it is satisfactory to find that it more than repays by the power which its improvement affords them.
354. We may divide our more important and fundamental instruments into four classes Those for measuring Time;
Space, linear or angular;
Mass. Other instruments, adapted for special purposes such as the measurement of Temperature, Light, Electric Currents, etc., will come more naturally under the head of the particular physical energies to whose measurement they are applicable. Descriptions of
self-recording instruments such as tide-gauges, and barometers, thermometers, electrometers, recording photographically or otherwise the continuously varying pressure, temperature, moisture, electric potential of the atmosphere, and magnetometers recording photographically the continuously varying direction and magnitude of the terrestrial magnetic force, must likewise be kept for their proper places in our work.
Calculating Machines have also important uses in assisting physical research in a great variety of ways. They belong to two classes:
1. Purely Arithmetical, dealing with integral numbers of units. All of this class are evolved from the primitive use of the calculuses or little stones for counters (from which are derived the very names calculation and “The Calculus "), through such mechanism as that of the Chinese Abacus, still serving its original purpose well in infant schools, up to the Arithmometer of Thomas of Colmar and the grand but partially realized conceptions of calculating machines by Babbage.
II. Continuous Calculating Machines. These are not only useful as auxiliaries for physical research but also involve important dynamical and kinematical principles belonging properly to our subject.
355. We shall now consider in order the more prominent instru. ments of each of these four classes, and some of their most important applications :
Clock, Chronometer, Chronoscope, Applications to Observation
and to self-registering Instruments.
Dividing Engine, Theodolite, Sextant or Circle.
Dynamometer. Among Standards we may mention1. Time.- Day, Hour, Minute, Second, sidereal and solar. 2. Space.-Yard and Métre: Radian, Degree, Minute, Second. 3. Force.—Weight of a Pound or Kilogramme, etc., in any par
ticular locality (gravitation unit); poundal or dyne, Kinetic
Unit 4. Mass.-Pound, Kilogramme, etc.
356. Although without instruments it is impossible to procure or apply any standard, yet, as 'without the standards no instrument could give us absolute measure, we may consider the standards firstreferring to the instruments as if we already knew their principles and applications.
357. First we may notice the standards or units of angular peasure:
Radian, or angle whose arc is equal to radius;
Degree, or ninetieth part of a right angle, and its successive subdivisions into sixtieths called Minutes, Seconds, Thirds, etc. The division of the right angle into 90 degrees is convenient because it makes the half-angle of an equilateral triangle (sin) an integral number (30) of degrees. It has long been universally adopted by all Europe. The decimal division of the right angle, decreed by the French Republic when it successfully introduced other more sweeping changes, utterly and deservedly failed.
The division of the degree into 60 minutes and of the minute into 60 seconds is not convenient; and tables of the circular functions for degrees and hundredths of the degree are much to be desired. Meantime, when reckoning to tenths of a degree suffices for the accuracy desired, in any case the ordinary tables suffice, as 6' is to of a degree.
The decimal system is exclusiveiy followed in reckoning by radians. The value of two right angles in this reckoning is 3:14159... , or T. Thus a radians is equal to 180°. Hence 1800 = 7 is 57°29578 ..., or 57° 17' 44":8 is equal to one radian. In mathematical analysis, angles are uniformly reckoned in terms of the radian.
358. The practical standard of time is the Siderial Day, being the period, nearly constant', of the earth’s rotation about its axis (S 237). From it is easily derived the Mean Solar Day, or the mean interval which elapses between successive passages of the sun across the meridian of any place. This is not so nearly as the former, an absolute or invariable unit; secular changes in the period of the earth's
? In our first edition of our larger treatise it was stated that Laplace had calculated from ancient observations of eclipses that the period of the earth's rotation about its axis had not altered by tooddooo of itself since 720 B.C. In § 830 it was pointed out that this conclusion is overthrown by farther information from Physical Astronomy acquired in the interval between the printing of the iwo sections, in virtue of a correction which Adams had made as early as 1863 upon Laplace's dynamical investigation of an acceleration of the moon's mean motion, produced by the Sun's attraction, showing that only about half of the observed acceleration of the moon's mean motion relatively to the angular velocity of the earth's rotation was accounted for by this cause. I Quoting from the first edition, $ 830.] “In 1859 Adams communicated to Delaunay his final result :-chat at *the end of a century the moon is 5'7 before the position she would have, "relatively to a meridian of the earth, according to the angular velocities of the "ewo motions, at the beginning of the century, and the acceleration of the "moon's motion truly calculated from the various disturbing causes then recog; "nized. Delaunay soon after verified this result: and about the beginning of *1866 suggested that the true explanation may be a retardation of the earth's wrotation by tidal friction. Using this hypothesis, and allowup for the conse*quent retardation of the moon's mean motion by tidal reaction ($ 276), Adams, in an estimate which he has communicated to us, founded on the rough as. 'sumption that the parts of the earth's retardation due to solar and lunar tides “are as the squares of the respective tide-generating forces, finds 22' as the error by which the earth would in a century get behind a perfect clock rated
at the beginning of the century. If the retardation of rate giving this integral "effect were uniform ($ 32), the earth, as a timekeeper, would be going slower
by '22 of a second per year in the middle, or '44 of a second per year at the "end, than at the beginning of a century."