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a result which depends upon no property whatever of the magnets employed, except M the moment of inertia of A. To the determination of the numerical value of M we will now give our attention.

If, as is usually the case with large magnets, the form of the magnet be a parallelopiped, and its structure mass in grains homogeneous, the value of M will be

12

X

{(length in feet)2 + (breadth in feet)}. This calculation is easily made with accuracy. It is necessary to add the moment of inertia of the stirrup or other hook which supports the magnet and vibrates with it: this in general is a very small quantity, and can be obtained with sufficient accuracy by weighing that apparatus and estimating its radius of gyration.

If the form of the magnet be not so simple, or if there be any grounds for suspicion of the accuracy of this process, the device proposed by Weber may be adopted. The time of vibration of the magnet having been observed, as above mentioned, the two ends of the magnet are then loaded with brass weights, very carefully weighed, which rest upon the magnet by sharp points, so that the weights do not partake of the cir

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cular movement of the magnet; and the distance between these points is measured. In that state, the time of vibration is again observed. As the force which causes the vibration is the same in both cases (namely, the action of terrestrial horizontal magnetism upon the magnet), the moments of inertia will be proportional to the squares of the two times of vibration. But the difference between the two moments of inertia is merely the moment of inertia of the two brass weights, each being supposed collected at its sharp supportingpoint, and admits of being accurately computed. Then the difference of the moments of inertia of the magnet in its two states being known, and their proportion being known, each of them is determined accurately.

Whichever method is used, the numerical value of Mis found and substituted in the expression for E, and the numerical value of E is obtained.

In the observation of deflexion described in Articles 25 to 29, it is evident that the comparison of the magnet-powers E and A implies that their numerical values are referred to the same unit. And in the investigation, in the present Article, of the measure of the Earth's action upon the magnet, we have used exactly the same formula as in Article 21, which is founded on the methods of preceding Articles, in which all are referred to the same unit. It follows that the numerical value which we have found for E is referred to the same unit: namely (see Article 29) to the magnet-power, or to the magnetic action at distance 1, of the standard magnet S or its equal S', which are such that, when

the distance of their centers is 1, the angular momentum produced in S" by S broadside-on (using the first term only of the formula of force), is 1.

In general, the two operations (deflexion and vibration) can be performed in so short a time that the effects of change of temperature and change in the Earth's force may be neglected. If it is thought necessary to recognise them, reference must be made, for temperature-correction, to the experiments of Article 31, and for the terrestrial change, to observations of minute changes (to be mentioned hereafter, Article 83.) Corrections for torsion-force of the suspension-thread ought to be applied on the principles of Article 25. There is also another small correction, for magnetism induced in the needle A by the Earth's action: for this we must refer to a succeeding section, Article 72.

33. Investigation of the proportion in which the numerical value for E will be altered, when, instead of using the foot and the grain as units, we use other units, as the millimètre and milligramme.

Let the foot =p millimètres, and the grain =q milligrammes. Suppose the observations of Articles 28 and 31 adopted without any modification of circumstances, and let us examine how the resulting formulæ will be modified by use of the new units. The experiment of Article 27 practically gives 24.c3 With the new units, sin

=

- E. sin ; or

E

=

2

A sin o

will not be altered, but c (numerically) will be p times

as great as before, and

E
A

will be (numerically) p3 times

as great as before. The experiment of Article 32 gives

EA =

4π2 Τ which depends on product of mass by square of distance from center of angular motion, will be (numerically) p’q

M; Tis not altered by the new units; but M,

E

times as great as before. The product of by EA will

Α

therefore be (numerically) times as great as before:

Ρ

and the numerical expression will be

great as before.

12 times as

The value of p is 304-7947: that of զ is 64 79895; therefore the new numerical expression for E on the Metric system will be formed by multiplying that on the English system by

6479895

304-7947

or 0.46108.

The same numerical result would have been obtained if the units employed, in the Metric system, had been the mètre and gramme.

34. Special values of E: historical physical change in the value: lines on the Earth's surface passing through points of equal horizontal force.

The mean value of E found at Greenwich in the year 1867 was 3.851 in English measure, or 1.776 in Metric measure. In 1848 its value at Greenwich, in English measure, was 3.722. The increase in 19 years

is in the proportion of 29 to 30: its rate of increase in successive years is sensibly uniform. We believe that this is the longest series of accurate determinations of horizontal force made in one place.

From all the comparative observations of horizontal force, made by the methods of Article 25, which could be collected about forty years ago, combined by the aid of a theory to which allusion is made in Article 24, (to be fully explained in Articles 47 and 49), Gauss formed a series of maps of lines of equal horizontal

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