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nature a periodic function of the independent variable. The harmonic analysis ($ 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 a denote an angle increasing in proportion to t, the time, through four right angles in the period, T, of the phenomenon; so that

27t
0=

T
let f(0)=A, +1, cos 0 + A, cos 20+ ...

+ B, sin 0+ B, sin 20+. where Ag, A4, Ag, ... Bu, B,,.

2,... are unknown co-efficients, to be determined so that f (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 co-efficients 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 co-efficients 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.

271 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 co-efficients, by the easiest and most direct elimination.

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CHAPTER IV.

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 purposes, 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;
Force;

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 emergencies to whose measurement they are applicable.

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355. We shall now consider in order the more prominent instruments of each of these four classes, and some of their most important applications :

Clock, Chronometer, Chronoscope, Applications to Observation

and to self-registering Instruments.
Vernier and Screw-Micrometer, Cathetometer, Spherometer,

Dividing Engine, Theodolite, Sextant or Circle.
Common Balance, Bifilar Balance, Torsion Balance, Pendulum,

Dynamometer.
Among Standards we may mention-

1. Time.-Day, Hour, Minute, Second, sidereal and solar.
2. Space.-Yard and Métre: Degree, Minute, Second.
3. Force.—Weight of a Pound or Kilogramme, etc., in any par-

ticular locality (gravitation unit); 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. We need do no more than mention the standard of angular measure, the Degree or ninetieth part of a right angle, and its successive subdivisions into sixtieths called Minutes, Seconds, Thirds, etc. This system of division is extremely inconvenient, but it has been so long universally adopted by all Europe, that the far preferable form, the decimal division of the right angle, decreed by the French Republic when it successfully introduced other more sweeping changes, utterly failed. Seconds, however, are generally divided into decimal parts.

The decimal division is employed, of course, when circular measure is adopted, the unit of circular measure being the angle subtended at the centre of any circle by an arc equal in length to the radius. Thus two right angles have the circular measure a or 3:14159..., so that 7 and 180° represent the same angle: and the unit angle, or the angle of which the arc is equal to radius, is 57°29578..., or 57° 17' 44":8. (Compare $ 41.) Hence the number of degrees n in any angle a given in circular

0 measure, or the converse, will be found at once by the equation

180' and therefore n=0 x 57°•29578...=0x57° 17' 44":8...

358. The practical standard of time is the Siderial Day, being the period, nearly constant, of the earth's rotation about its axis (§ 237). It has been calculated from ancient observations of eclipses that this. has not altered by

of its length from 720 B.C. : but an error has been found in this calculation, and the corrected result renders it probable that the time of the earth's rotation is longer

n

TT

1 10,000,000

1 2,700,000

by

now than at that date. 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 nct so nearly as the former, an absolute or invariable unit; secular changes in the period of the earth’s revolution round the sun affect it, though very slightly. It is divided into 24 hours, and the hour, like the degree, is subdivided into successive sixtieths, called minutes and seconds. The usual subdivision of seconds is decimal.

It is well to observe that seconds and minutes of time are distinguished from those of angular measure by notation. Thus we have for time 13h 43m 278.58, but for angular measure 13° 43' 27":58.

When long periods of time are to be measured, the mean solar year, consisting of 366.242203 siderial days, or 365•242242 mean solar days, or the century consisting of 100 such* years, may be conveniently employed as the unit.

359. The ultimate standard of accurate chronometry must (if the human race live on the earth for a few million years) be founded on the physical properties of some body of more constant character than the earth: for instance, a carefully-arranged metallic spring, hermetically sealed in an exhausted glass vessel. The time of vibration of such a spring would be necessarily more constant from day to day than that of the balance-spring of the best possible chronometer, disturbed as this is by the train of mechanism with which it is connected: and it would certainly be more constant from age to age than the time of rotation of the earth, retarded as it now is by tidal resistance to an extent that becomes very sensible in 2000 years; and cooling and shrinking to an extent that must produce a very considerable effect on its time-keeping in fifty million years.

360. The British standard of length is the Imperial Yard, defined as the distance between two marks on a certain metallic bar, preserved in the Tower of London, when the whole has a temperature of 60° Fahrenheit. It was not directly derived from any fixed quantity in nature, although some important relations with natural elements have been measured with great accuracy. It has been carefully compared with the length of a second's pendulum vibrating at a certain station in the neighbourhood of London, so that if it should again be destroyed, as it was at the burning of the Houses of Parliament in 1834, and should all exact copies of it, of which several are preserved in various places, be also lost, it can be restored by pendulum observations. A less accurate, but still (unless in the event of earthquake disturbance) a very good, means of reproducing it exists in the measured base-lines of the Ordnance Survey, and the thence calculated distances between definite stations in the British Islands, which have been ascertained in terms of it with a degree of accuracy sometimes within an inch per mile, that is to say, within about

361. In scientific investigations, we endeavour as much as possible to keep to one unit at a time, and the foot, which is defined to be one-third part of the yard, is, for British measurement, generally

60000

adopted. Unfortunately the inch, or one-twelfth of a foot, must sometimes be used, but it is subdivided decimally. The statute mile, or 1760 yards, is unfortunately often used when great lengths on land are considered; but the sea-mile, or average minute of latitude, is much to be preferred. Thus it appears that the British measurement of length is more inconvenient in its several denominations than the European measurement of time, or angles.

362. A far more perfect metrical system than the British, is the French, in which the decimal division is exclusively employed. Here the standard is the Mètre, defined originally as the ten-millionth part of the length of the quadrant of the earth's meridian from the pole to the equator; but now defined practically by the accurate standard mètres laid up in various national repositories in Europe. It is somewhat longer than the yard, as the following Table shows. . Its great convenience is the decimal division. Thus in any expression the units represent mètres, the tens decamètres, etc.; the first decimal place represents decimètres, the second centimètres, the third millimètres, and so on.

Inch=25.39954 millimètres. Millimètre=:03937079 inch. Foot=3.047945 decimètres. Decimètre=:3280899 foot. British land-mile = 1609:315 Kilomètre=-6213824 landmètres.

mile. Sea-mile=1851.851 mètres. *Kilomètre=:54 sea-mile.

363. The unit of superficial measure is in Britain the square yard, in France the mètre carré. Of course we may use square inches, feet, or miles, as also square millimètres, kilomètres, etc., or the Hectare=10,000 square mètres.

Square inch = 6°451367 square centimètres.
foot = 9.28997

decimètres.
yard = 83.60971 decimètres.
Acre

= .4046711 of a hectare. Square mile

= 258.9895 hectares. Hectare = 2:471143 acres. 364. Similar remarks apply to the cubic measure in the two countries, and we have the following Table :

Cubic inch = 16.38618 cubic centimètres.

foot = 28.315312 decimètres, or litres. Gallon = 4.54346 litres.

= 277.274 cubic inches. Litre = 0.035317 cubic feet. 365. The British unit of mass is the Pound (defined by standards only); the French is the Kilogramme, defined originally as a litre of water at its temperature of maximum density; but now practically defined by existing standards. Grain = 64:79896 milligrammes. Gramme =15°43235 grains. Pound 453.5927 grammes.

Kilogram. = 2.20362125 lbs. Professor W. H. Miller finds (Phil. Trans., 1857) that the 'kilo

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