In this formula, A is a constant depending only on the nature

B

of the steel, and ẞ a quantity of the form, B being a second constant. The diameter of the needles is represented by a.

The formula is conformable to the results of Coulomb's experiments*. I propose to verify it for the particular case of needles of very slight diameter and little length.

1. Needles of the same diameter.-The temper of the needles compared should be identical; the breaking of a saturated, quite regular cylindrical needle furnishes saturated needles in that condition. They are saturated; for the primitive needle possesses in every point of it a higher degree of magnetism than corresponds to the saturation of the fragment which belongs to it after the breaking. The breaking itself has, according to the preceding, no other effect than that which would result from the separation of the parts which were in precise juxtaposition. Finally, the temper is as identical as possible; we shall even see that employing the results eliminates the minute differences of local temper which cannot be entirely avoided.

The mother needle is magnetized to saturation, by at least four Bunsen elements, in a coil of 25 centims. length, very regular, and formed of three superposed layers of wire. In order to make an experiment, the length and magnetic moment of the entire needle are measured; the two ends are then removed by breaking them off at 3 or 4 centims. distance from the extremities, preserving the end fragments; the length of the middle fragment and its magnetic moment are measured, and by successive breakings, each accompanied by two measurements, it is reduced to a length of 1 or 2 millims. The experiment is finished by measuring the length and magnetic moment of the two ends, entire and reduced to successively shorter lengths.

It is rare that all the measurements can be accomplished with one and the same apparatus; two are therefore employed :-one less delicate, for the greater lengths; the other more delicate, for the shorter ones. The ratio of the two apparatus is determined with the greatest care by a number of measurements taken from both.

In order to be clear of all accidental irregularities, the results of the experiments are represented by a curve-the lengths x of the needles being taken for the abscissæ, and the corresponding magnetic moments for the ordinates. This curve is traced with extreme care by means of the measurements made on the middle fragments of the mother needle. If this needle is saturated, the

* Coulomb, "Détermination des forces qui ramènent différentes aiguilles aimantées à leur méridien magnétique,” Mém. de l'Institut, vol. iii.

points characteristic of the end fragments and of the entire needle will fall of themselves on the curve. If this condition is not satisfied, the experiment will be rejected.

Experiment shows that different fragments of the same needle broken before magnetization, magnetized separately to saturation, give points that place themselves on the curve traced fro the breaking of one of them (the longest, for instance). This important experiment proves that in the present case the breaking has really no effect at all.

Equation (1) represents a curve tangent to the axis of x at the origin, and an asymptote to the right line

The curves representing the experiments present the same general characters. To make the comparison, the asymptote of the experimental curve is determined with the utmost care. In fact, starting from a length of from 10 to 40 centims. according to the diameter, the points characteristic of the needles fall rigorously in a right line, or only deviate within the limits of errors of experiment; the asymptote is therefore perfectly determined. Let D be its abscissa at the origin, C its angular coefficient; the equation can be put in the form

This formula has served for calculating the magnetic moment of short needles; the real moment is determined directly upon the experimental curve.

It is in this way that the following Tables have been formed. The first column contains the lengths of the needles; the second, the observed magnetic moments in arbitrary units; the third, the moments calculated by the formula (1'); the last two, the absolute and relative differences of the observed from the calculated moment.

The experiments were made on needles of 0.175, 0.282, 0·368, and 0.551 millim. diameter. We will confine ourselves to the results furnished by the last three, because the representative curve of the first is too near a right line, for all lengths above 2 millims., for any certain conclusions to be deduced relative to the part of the curve in the vicinity of the origin.

Beyond 10 millims. for the thickest needle, and 6 millims. for the thinnest, the characteristic points are, theoretically and practically, confused with the asymptote.

The agreement of calculation and experiment is very remarkable for the needles which are not too short in proportion to their diameter. It was for this case only that Green established the formula which we are engaged in verifying. For extremely short needles, in all the experiments the observed are invariably greater than the calculated numbers. The absolute differences are, it is true, very small; but they exceed the limit of errors of observation, and as much more as the diameter of the needles is more considerable. Nevertheless they are not sufficiently great to permit us to seek empirically the form of the correction which would have to be added to the formula to make it perfectly

accurate.

2. Needles of different diameters. For needles of different diameters Green's formula admits of other verifications. The angular coefficient C, of the asymptote, has to be proportional to the square of the diameter of the needles, and the abscissa at the origin, D, proportional to their diameter.

It is easy to attach a physical meaning to the quantities C and D. Let us consider two needles of the same diameter, sufficiently long for their characteristic points to place themselves sensibly on the asymptote. Their magnetic moments y and y' are represented by the corresponding ordinates of the asymptote; that is to say, we have

On the other hand, we know, from Coulomb, that in long needles the distance of the poles from the extremities is constant, whatever the length may be. Let P be that distance, and μ the quantity of magnetism of each pole (also constant); we have

The systems (3) and (4) are incompatible, unless we have at the same time C=μ and D=2P; so that the semiabscissa at the origin of the asymptote is equal to the distance of the needle's pole from the corresponding extremity, and the angular coefficient of the same right line is equal to the quantity of magnetism of each pole.

Thus, in the case of long cylindrical needles of different diameters, Green's formula expresses the proportionality of the power of the poles to the square of the diameter, and the pro

portionality of their distance from the ends of the needle to the first power of the diameter.

Comparison of the results furnished by observation on needles of different diameters is attended with great practical difficulties. The multiplicity of comparisons of apparatus required by these experiments, the considerable influence of the slightest errors in the estimation of the minute diameters upon the ratios to be determined, and, above all, the difficulty of giving an identical temper to needles of different diameters are grave obstacles which it is not easy entirely to surmount. It is right, however, to remark that the difficulty in regard to the tempering is less for needles that are tempered very hard, such as we have always employed, because in this case the coercive power varies little for rather large variations in the temperature at which the steeping was effected.

The following Table relates to the law of the polar distances. The first column gives the diameters of the needles, the second the abscissa D, the third the value of the ratio

D

d

(which should

be constant); and the last column gives the differences of the numbers in the third column from their mean.

These numbers verify the law, if we take into consideration the above-mentioned multiplicity of the causes of error.

The law relating to the power of the poles is equally well verified, as will be seen in Table V., the third and fourth columns of which give the absolute and relative values of the difference

give

C

between and the corresponding mean.

d2