Magnetism and dimension-ratio.-Of equally great interest, moreover, is the other phase of these phenomena in which magnetizability is regarded as a function of the independent variable, the dimension-ratio, whereas hardness appears merely as an arbitrary constant. We will endeavor in this place to give an account of our data, sufficiently detailed to admit satisfactorily of a comparison with results of earlier observers. In the absence of a better means of estimating hardness the oxide tints were largely made use of, and the wires examined, therefore, classified as hard, yellow annealed, blue annealed, and soft. In conformity with this method of description, we will attempt to give the parameter (hardness) in our own results, such values as will correspond with the said four States. The following special experiments furnish ap proximate data for the expression of the degree of hardness corresponding to any given oxide tint in our own galvanic scale:

It is well to note that the logarithmic interval between yellow annealed and blue annealed is about equivalent to the same interval between blue annealed and soft, and about twice as large as that between glass-hard and yellow annealed. Referred to our method of tempering, the values for yellow annealed and blue annealed would agree very nearly with three hours of annealing in aniline vapor (mean s=26.7 and 26.0) and one minute of annealing in melting lead (mean s=19.5 and 19.7), respectively.

The smaller galvanic values for these states correspond to the smaller values for the soft state (mean s=15.3 and 15.1), respectively, and the difference is, therefore, obviously due to carburation. Hence for our present purposes it is fully sufficient to select from tables 62 and 63 the following mean values of magnetism for the degrees of hardness in question:

TABLE 64.-Thick wire.

Graphic representation.-These data enable us to construct the curves in Figs. 19 and 20.

The curve glass-hard is concave as regards the axis of abscissæ throughout. Rising very rapidly at first it finally ascends to a distinct limiting value or horizontal asymptote. The curve soft, on the other hand, rises very slowly in its earlier stages, and is convex as regards the axis of abscissæ. From here it passes rapidly upward through a point of circumflexion into concavity towards X, and then above the former curve. Finally the rate of ascent again decreases so that a horizontal asymptote must eventually also be reached, but apparently at a later stage of the progress than is the case with hard wire. From either of these two curves, which describe the variations of the extreme states, we may, by the process of annealing, pass continuously to the other; but the manner of such passage from the one to the other, resulting from a continuous change of parameter (hardness), is excessively complicated. Incipient annealing of a glass-hard rod produces a distinct though small descent of the original curve as a whole. As annealing progresses, the farther end of the curve is always the first to rise and pass above the original curve in such a way that the point of intersection of the new curve and the original curve, glasshard, moves rapidly from greater to smaller values of the dimensionratio. When the curve "yellow annealed" is reached the part between a=14 or 18, respectively, and a∞o has already been elevated above the curve glass-hard. In the case of the "blue annealed" curve we observe the part between a=15 or 20, respectively, and a=∞ to have risen enormously; but on the other hand the part of this curve between a= 0 and a=15 or 20, respectively, having descended gradually, is now distinctly convex as regards the axis of abscissæ. Passing from blueannealed to soft, we find the left-hand part of the curve falling at a more rapid rate, while the right-hand branch still rises slowly, reaching a superior limit and then falling rapidly into coincidence with the curve for the soft state, already discussed.

We have remarked that the phenomenon of variation of magnetizability, regarded as a whole, does not by any means seem to present a certain critical value for the dimension-ratio, below which magnets behave differently than above it. If only the four curves hard, yellow annealed, blue annealed and soft were known, as they appear in Figs. 19 and 20, there would appear to be reasons for considering a=15 or 20, respectively, as a critical value of this kind; for it is here that the three curves approximately intersect each other. How unjustifiable, however, such conclusions are, will be obvious from a mere glance at Figs. 17 and 18, in which the curves for this value of a manifest no change of character whatever.

Magnetism and density.-We may remark here that when the rod is very long in comparison with its diameter, its specific magnetism (saturation presupposed) becomes constant or independent of the dimensions.

In Green's equation for the distribution of magnetism in a cylinder of finite length, 21, and radius r.

where A is the linear density of free magnetism at a distance x from the middle of the cylinder, K the coefficient of magnetization, X the magnetizing force, p a numerical quantity related to K; let KX be accepted as the expression for coercive force, F, and put for greater simplicity

into a constant value, depending only on the material. Biot 132 put p = 0.124, approximately.

Now it is remarkable that in this case, in which moreover the expres sion for the linear distribution of magnetism attains its simplest form, we are led to infer that the (linear) rods are capable of retaining more magnetism permanently in proportion as they are softer. But this general deduction will not hold good as far as the soft state. Figure 20 gives as yet no decisive answer to this question. However, in passing from the blue annealed to the soft condition, steel passes through a state of maximum density; and our results thus give warrant to the surmise that the greatest attainable magnetization can be imparted to steel, when the hardness of the necessarily linear rod has that particular value which is characterized by maximum density. This digression suggests itself naturally here, but will be made the subject of a special inquiry. From figure 17, however, it is already quite apparent that the said unique maximum will be considerably larger than 785 c. g. s. units of intensity (magnetic moment per unit of volume), the value thus far assumed and derived from an incidental result of Kohlrausch.133

132 Biot: Traité de Phys., III, p. 76, 1816.

133 Cf. Gordon: Electricity and Magnetism, I, p. 156, London, 1880.

(737)

Earlier results interpreted.-It will now be in place to compare our results with those of Ruths. We will select from his data the series contained on p. 47 of his memoir, as these may be regarded typical. Here again, those alone can be considered which refer to magnets as nearly as possible in the permanently saturated state. In the place of the absolute magnetic moment of his steel rods we have deduced the corresponding values for specific magnetism from his data. This greatly facilitates comparison.

The first comment to be made in this connection has reference to our remarks given in p. 117, on the structural influence of tempering. Ruths' magnets were all of the same lengths, whereas the thickness varied from 0.2 centimeters to 0.6 centimeters. To impart like degrees or similar states of hardness to rods varying in thickness to this very large extent, is manifestly impossible, even if the material were throughout perfectly identical. It follows that the divers values of specific magnetism obtained cannot even be compared one with another. Hence the attempt to arrive at the nature of the functionality between permanent magnetism and ratio of dimensions from these results, must necessarily be futile. Ruths' data are unavailable, therefore, for the construction of curves, corresponding to those in figure 19. We infer from Ruths' table that in general the specific magnetism increases with the dimension ratio. In the case of the thicker magnets (IV, V, VI) Ruths found values for specific magnetism much larger than our own. In accordance with these, the intersection of the curves glass-hard and blue annealed would occur for a=30; the corresponding point for blue annealed and yellow annealed lying at a=28. In accordance with Ruths' results, moreover, figure 19 would have to be changed in such a way that the curve glass-hard (to agree with Ruths' large values) rise at a more rapid rate, thus falling short of intersection with the other two curves until these have intersected each other at about a=20. Then its passage is through blue annealed at about a 25, and through yellow annealed at about a=40, where the latter curve is also supposed to rise more rapidly than is the case in our results. But with this apparently satisfactory interpretation, the equiv alence of blue annealed and yellow annealed for a=50, is wholly in discordance.

Fromme uses eight rods, all of the same length, four of which are of