application.

least squares.

to the smallest probable error. As an instance--by the prin- Practical
ciples of physical astronomy, the place of a planet is calculated Method of
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 be the calculated R.A. of a planet: da, de, d, etc.,
the corrections required for the assumed elements-the true
R.A. is
0+Ada+Ede+ПId≈+etc.,

where A, E, II, etc., are approximately known. Suppose the
observed R.A. to be, then

a known quantity, subject to error of observation. Every obser-
vation made gives us an equation of the same form as this, and
in general the number of observations greatly exceeds that of the
quantities da, de, da, etc., to be found. But it will be sufficient to
consider the simple case where only one quantity is to be found.
Suppose a number of observations, of the same quantity x, lead
to the following equations :-

x=B1, x=B2, etc.,

and let the probable errors be E1, E.,... Multiply the terms of
each equation by numbers inversely proportional to E1, E, ....
This will make the probable errors of the second members of all
the equations the same, e suppose. The equations have now the
general form
ax=b,

and it is required to find a system of linear factors, by which
these equations, being multiplied in order and added, shall lead
to a final equation giving the value of x with the probable error a
minimum. Let them be fi, f2, etc. Then the final equation is

by the theorems of § 393, if P denote the probable error of x.

Method of

least squares.

Methods of representing

results.

Hence

Σ(ƒ3) (af)3

is a minimum, and its differential coefficients

with respect to each separate factor f must vanish.

This gives a series of equations, whose general form is

ƒΣ(af)—aΣ(ƒ2)=0,

which give evidently f1=a1, f1=a1, etc.

Hence the following rule, which may easily be seen to hold for any number of linear equations containing 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 coefficient 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.

That is, in the simple case taken above,

(ax-b) minimum.

For it is evident that this gives, on differentiating with respect
Za(ax-b)=0,

to x,

which is the law above laid down for the formation of the single equation.

395. When a series of observations of the same quantity experimental 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, that we 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.

396. Thus if the abscissæ represent intervals of time, and

the ordinates the corresponding height of the barometer, we Curves. 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.

397. 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, liberâ 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 underground were found by Forbes sufficient for a very accurate approximation to the law of the phenomenon.

tion and

formulæ.

398. As an instance of the processes employed for obtaining Interpolaan empirical formula, we may mention methods of Interpola- empirical tion, 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 formula 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

y=f(x。+x−x ̧)=f(x)+(x−xo)ƒ'(x)+'

(x-xo).
1.2

ƒ"(x)+...

Interpolation and empirical formulæ.

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.

In finite differences we have

f(x+h)=D'f'(x)=(1+A)'f'(x)

12 1.2

=ƒ{(x)+h\f(x)+ "',A2ƒ{(x)+.....

(2).

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, X2, xn a function takes the values y1, Y1⁄2, Ys, yn, Lagrange gives for it the obvious expression

Y31

Here it is of course assumed that the function required is a rational and integral one in x of the n-1th degree; and, in 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 ƒ ́(x。), f(x), ... in the first, or of Af(x), Af(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 f(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 7 of a rod of metal as depending on its temperature t, we may assume from (1)

1=1+A(t-to)+B(t-to),
1, being the measured length at any temperature fo

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.

401. We may divide our more important and fundamental instruments into four classes—