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 D a the abscissa D, the third the value of the ratio (which should be constant); and the last column gives the differences of the numbers in the third column from their mean. TABLE IV. d. millim. 0.175 0-282 0:368 0:551 1.036 1.290 1.988 D. millim. 1.77 2:45 3.60 5.25 9.90 12:40 16 80 10-113 9.783 +0.388 9.528 +0.133 9.556 9.612 8.451 +0.161 +0.307 -0.944 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 С between 22 and the corresponding mean. and the other factor d. D= +3·094 -4.067 -0.094 +1.070 In short, experiment confirms in a very remarkable manner the various laws contained in Green's formula, except in the case of excessively short needles. It follows that, with the exception mentioned, the distribution of free magnetism in the needles is also represented by the formula given for it by Green, of which our formula is a consequence*. The quantity of free magnetism in a section perpendicular to the axis, of thickness da, situated at a distance from the middle, is, according to that formula, Aa B A being half the length of the needle. This formula is equivalent to Biot's, and faithfully represents the results of Coulomb's experiments. Poles of short needles.-The magnetic moment of a needle can always be considered as the product of two factors, one of which represents the distance of the poles, the other the quantity of magnetism of each of them. According to Green's formula, the distance of the poles is 2 eBλ-e-Bλ В ева те-ва 2 ева те-ва r. 0.055 0.073 0.002 0.019 * The pole is defined as the projection of the centre of gravity of the curve of distribution r=Aa2ß on the magnetic axis of the needle. The distance D of the two poles has been calculated according to this definition. Phil. Mag. S. 4. Vol. 49. No. 324. March 1875. 0 2 and Aa2 respectively for very B These factors reduce to 2λ high values of λ. On the other hand, the ordinate y of a curve may always be regarded as the product of the angular coefficient of the tangent at the point considered, into the difference of the abscissa 2λ and the abscissa at the origin of the tangent. From these principles the magnetic moment of the needle considered is represented by the product of the two factors 2λ 1 2 eßλ-e-Bλ В ева те-ва 2 + βλ 2 +e-Bλ ρβλ A a2 2' 2 (8) B These factors, like the preceding, reduce to 2λ- and Aa2 for very high values of λ. Hence it is that we have before employed the asymptote for determining the polar distance of long needles; but this cannot be extended to the tangents, as I at first expected, since the expressions (8) differ from expressions (6) and (7) by the variable factor 1+ The method eBλ +e-Bλ° which has served us for the determination of the magnetic moments cannot, therefore, enlighten us on the independent variation of the two factors on which they depend. 2 3. Breaking of non-saturated Needles perpendicular to the axis. A. Regular needles. When a needle is magnetized regularly without being saturated, a sufficient length of its extremities can be removed, and then the middle portion can be treated like that thus taken from a saturated needle. A characteristic curve is obtained tangent to the axis of x at the origin, and presenting an asymptote the curve of which converges rapidly. The semiabscissa at the origin of this asymptote is the distance of the poles of long rupture-needles (obtained by breaking) from their extremities*. This distance is therefore constant in rupture-needles as in saturated needles. Moreover two cases are to be distinguished : 1. When the needles are thin (0-175 to 0.551 millim. diameter) the distance of the poles from the extremities of ruptureneedles depends only on the diameter, and is the same as in satu* Demonstration the same as for saturated needles. 2 These factors reduce to 2λ-and Aa2 respectively for very β bigh values of X. On the other hand, the ordinate y of a curve may always be regarded as the product of the angular coefficient of the tangent at the point considered, into the difference of the abscissa 2 and the abscissa at the origin of the tangent. From these principles the magnetic moment of the needle considered is represented by the product of the two factors 2 ABA+e-BA 2 eBA + e-Bλ) (8) 2 B These factors, like the preceding, reduce to 21- and Aa for very high values of λ. Hence it is that we have before employed the asymptote for determining the polar distance of long needles; but this cannot be extended to the tangents, as I at first expected, since the expressions (8) differ from expressions The method (6) and (7) by the variable factor 1+ 2 which has served us for the determination of the magnetic moments cannot, therefore, enlighten us on the independent variation of the two factors on which they depend. 3. Breaking of non-saturated Needles perpendicular to the axis. A. Regular needles. When a needle is magnetized regularly without being saturated, a sufficient length of its extremities can be removed, and then the middle portion can be treated like that thus taken from a saturated needle. A characteristic curve is obtained tangent to the axis of a at the origin, and presenting an asymptote the curve of which converges rapidly. The semiabscissa at the origin of this asymptote is the distance of the poles of long rupture-needles (obtained by breaking) from their extremities *. This distance is therefore constant in rupture-needles as in saturated needles. Moreover two cases are to be distinguished: 1. When the needles are thin (0-175 to 0.551 millim. diameter) the distance of the poles from the extremities of ruptureneedles depends only on the diameter, and is the same as in satu* Demonstration the same as for saturated needles. rated needles. Indeed the asymptotes of all the curves corresponding to needles of the same diameter cut the axis of x in exactly the same point. This has been verified On 3 curves for needles of 0.551 millim. 2 3 0.398 2 در در در millim. 3 9 10 y=mAa2 x Curve S. 5.60 12.00 20.00 30.00 42.00 55.50 70.20 85.60 99 Further, in the case we are considering, the complete curve is only a proportional reduction of the curve for saturated needles. It is exactly represented, within the same limits, by the equation (9). in which m is a factor whose value is less than 1, depending on the degree of magnetization of the mother needle. TABLE VI. e2+ e 2 Curve R. Observed. 4:40 8.60 14.15 21.00 29.50 39.60 βε 2 50.50 61.20 Calculated. در 4.013 8.597 14.317 21-492 30.088 39.760 50.290 61.323 The above Table refers to a needle of 0·551 millim. diameter. The second column contains the moment of saturation of the needles as furnished by the experimental curve, and the third the moment of the rupture-needles; the numbers in the fourth column were obtained by multiplying those in the second by the ratio m of the angular coefficients of the two asymptotes; the fifth column gives the differences between the observed and the calculated numbers. Above 10 millims. the curves approach very closely their asymptotes, and the comparison which forms the object of this Table ceases to be of interest. Note that the poles of short rupture-needles are situated the same as if the needles were saturated. II. These different results do not apply to thicker needles (of 1 to 2 millims. diameter for example). In the first place, the asymptotes to the different curves corresponding to needles of 02 the same diameter do not meet the axis of r at exactly the same point, but in points nearer to the origin in proportion as the degree of magnetization of the mother needle is less. Besides, the curves themselves are not proportional reductions of the same curve; and if we take the ratio of the magnetic moment of a rupture-needle to the corresponding saturated needle, this ratio approaches towards unity in proportion as the length of the needle diminishes. Therefore, if needles be taken from the middle of a non-saturated needle, but which is regular and from 1 to 2 millims. in diameter, the shorter they are the nearer are they to saturation. The difference which we have noticed in this respect between very thin and thicker needles is important for the theory of magnetism. It remains to examine the condition of the end fragments of non-saturated needles. I have limited myself to comparing the magnetic moment of these fragments with that of equal fragments from the middle of the needle. I ascertained that the moment of the end fragments is below that of the middle ones, and as much more so (1) As the primitive needle is shorter, (2) As the magnetization is less intense, (3) As the absolute length of the fragments is less. The following are some examples : Divided first into three, then into six equal fragments. Ratio of the end third parts to the middle ones sixth parts II. 0.803 0.646 millim. 141 2 Divided into five, then into ten and into twenty equal fragments. In the same experiment, the ratio of the twentieths occupying the second place from the ends, to the middle twentieths, was found to be 0.785. B. Needles presenting consequent points.-We have just seen that, in a regular needle, the fragments from the extremities possess a lower magnetic moment than those which come from the centre. The consequent points behave like poles of less force than the extreme poles; the fragments which include them possess, with equal length, a higher magnetic moment than that of the end fragments, but lower than that of the fragments derived from the interval between two consequent points. We conclude this section by indicating a means of verifying the perfect regularity of a magnetized needle :-After separating a sufficient length of the extremities, we break the middle piece into fragments of arbitrary, unequal lengths. If the mother needle was regular, the characteristic points obtained by taking for abscissa the length of the fragments, and for ordinate their magnetic moment, will be situated on a regular curve. The slightest irregularities will then be seen by simple inspection of the figure obtained. 4. Separation, parallel to the axis, of Prismatic Bundles.Observations on the Temporary Magnetism of Steel, The difficulty of breaking a needle along a plane parallel to its axis induced us to investigate the more practical case of superposed strips composing a bundle. The results furnished by this examination apply only approximately to breaking, since the latter may be regarded as the extreme case of separationwhen the parts facing each other on both sides of the plane of separation are at an indefinitely minute distance. We form a prismatic bundle with a number of pieces of watch |