Curves.

Interpola

tion and empirical formulæ.

24 feet underground were found by Forbes sufficient for a very accurate approximation to the law of the phenomenon.

398. 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 differential calculus, and the calculus of finite differences, give us formulæ for any required data; and Lagrange has shown how to obtain a very useful one by elementary algebra.

By Taylor's Theorem, if y=f(x), we have

where is a proper fraction, and x, is any quantity whatever. This formula is useful only when the successive derived values of f(x) diminish very rapidly.

a very useful formula when the higher differences are small.
(1) suggests the proper form for the required expression, but it
is only in rare cases that f(x), f" (x), etc., are derivable directly
from observation. But (2) is useful, inasmuch as the successive
differences, Af (x), Af(x), 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, x ̧, x a function takes the values y1, Y, Ya,... Y, Lagrange gives for it the obvious expression

1

+

Y2

1

+

tion and

formulæ.

455 Here it is of course assumed that the function required is a Interpolarational and integral one in x of the n-1th degree; and, in empirical general, a similar limitation is in practice applied to the other formulæ above; for in order to find the complete expression for f(x) in either, it is necessary to determine the values of f'(x), f" (x), ... in the first, or of Af (x), A°f (x), ... in the second. If n of the coefficients 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 ƒ(x), and the expressions become determinate and of the n-1th degree in x-x, and h respectively.

In practice it is usually sufficient to employ at most three terms of either of the first two series. Thus to express the length of a rod of metal as depending on its temperature t, we may assume from (1)

7 being the measured length at any temperature t„.

398'. These formulæ 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 formulæ 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.

functions.

In a large class of investigations the observed element is in Periodic its nature a periodic function of the independent variable. The harmonic analysis (§ 77) is suitable for all such. When the values of the independent variable for which the element has been observed are not equidifferent the coefficients, 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 0 denote an angle increasing in proportion to t, the time, through four right angles in the period, T, of the phenomenon; so that

Periodic functions.

let

f(0)=A+A, cos 0 + A, cos 20 + .....

...

1

2

where A, A, A,, B1, B,... are unknown coefficients, to be determined so that ƒ(0) 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 formulæ 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 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 coefficients, by the easiest and most direct elimination.

2π

CHAPTER IV.

MEASURES AND INSTRUMENTS.

of accurate

ments.

399. HAVING seen in the preceding chapter that for the Necessity investigation of the laws of nature we must carefully watch measureexperiments, 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 purposes, and the various standards or units which are employed in them.

400. 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, confusionslight, 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.

Classes of instru.

ments.

401. We may divide our more important and fundamental

instruments into four classes

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 :

I. 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. As these are not only useful as auxiliaries for physical research but also involve dynamical and kinematical principles belonging properly to our subject, some of them have been described in the Appendix to this Chapter, from which dynamical illustrations will be taken in our chapters on Statics and Kinetics.