of hardness and with the ratio of length and diameter. In place of a diagram in three dimensions, however, it is expedient to regard the one or the other variable, as an arbitrary constant, while the other passes through all possible values. We thus represent all relations by plane diagrams. Our results are compared in this way. It is, however, convenient to replace Tables 54 and 55 by others satisfactorily deducible from them by graphic interpolation, since the ratio of dimensions in the earlier experiments (Tables 54 and 55) do not compare well with those in the later (Tables 56 to 59). In other words, without such reconstruction of the former results, the difference of magnetic behavior of the two kinds of steel employed cannot be readily discerned. For this reason the Tables 60 and 61, following, have been prepared. The data contained are such as would have been found if the ratio of dimensions had been a=20, 40, 60, 80, and 100, respectively, in place of the values which actually obtained.

General inferences.-A detailed comparison of the results expressed in Tables 60 and 61 with those in Tables 56, 57, 58, 59, and all of these among themselves, shows conclusively that the magnetic behavior of

each particular wire is of a distinct and individual character. More perspicuously, even, the same fact is observable in a graphic representation of the data. The curves obtained for the several wires are by no means coincident. The thicker wire shows greater magnetizability than the wire of smaller diameter. In the case of both kinds of steel, galvanically hard rods are magnetizable to a lesser degree than soft rods. Moreover, there appears to exist a relation between the density and maximum permanent moment, in so far as the latter in general increases directly with the former. This effect is particularly apparent for the thicker wire. But the fundamental inference is this, that the relation between specific magnetism and hardness expressed galvanically, as exhibited by all the wires, is essentially of the same kind.

Magnetism and hardness.-The general contours of the families of curves expressing the functionality between magnetic moment, per unit of mass, hardness, ratio of dimensions, are of very great interest. It appears clearly that the latter variable by no means deserves the exclusive importance which has hitherto been attributed to it, at least not in the accepted sense that a certain critical value exists (Ruths, for instance, puts it, a=35) below which magnets differ in a pronounced way as regards magnetizability, from magnets with a larger dimension-ratio. The variations of the series of curves belonging to magnets taken from one and the same wire are obviously of like nature; in other words, obtained from a definite law by the mere change of a parametric constant, a. In general, all magnets, whether long or short, after incipient annealing diminish in magnetizability, until a well-marked minimum of this quality is reached. If thereafter annealing be continued magnetizability again increases, attains a maximum, and finally decreases again to the value for the soft state.

The causes which led earlier observers into partially erroneous in-ferences are not far to seek. With the ordinary very rough method of estimating the degree of hardness of steel by the aid of the oxide tints, the character of the curves, as a whole, did not appear. Hitherto data for only four points for each rod, corresponding to the hard, the yellow, the blue, and the soft states, were available. Irrespective of the vagueness of degrees of hardness thus defined it will readily be seen that where the full contours of the curves are unknown the grouping of these points is such as to suggest the said imperfect inference of a critical dimension-ratio.

The effect of a variation of dimensions is particularly apparent in its bearing on the position and value of the minima and maxima of the permanently saturated magnetic state. In proportion as the length of the magnets, or more accurately the ratio of dimensions increases, these singular points, which are nearly coincident in very short rods, move apart, the minimum occurring in the glass-hard state, the maximum passing at a very rapid rate toward the soft state. At the same time the maximum increases enormously in value, and for long, thin rods is

greater than twice the equivalent of the permanently saturated magnetic moment of the same rod when hard. As we approach short rods, as will be seen, the maximum flattens out, moving toward the minimum at a more rapid rate than the minimum progresses in the opposite direction. But the variations are such that for very short magnets, i. e., such for which the ratio of dimensions is below 5, the minimum will probably lose its marked character altogether, and eventually come into full coincidence with the maximum. Here, therefore, a uniformly continuous decrease of magnetizability from the glass-hard to the soft states is to be anticipated.

Graphic representation.-All these results may be made to appear with striking clearness by representing them graphically. It will be sufficient for this purpose to accept the mean of the two series of results for the thin sample of steel wire, and then to compare this with the mean or combined results (four series) obtaining for the thicker sample of wire. These mean values are given in the following tables, 62 and 63: TABLE 62.-Progressive states of magnetizability, varying with hardness.

TABLE 63.-Progressive states of magnetizability, varying with hardness.

On the basis of these tables the two families of curves, Figs. 17 and

18, have been constructed.

Hardness decreasing

a 20

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: