ε

and this is the ratio of the values of Q", at the ends of the helix. When b is 12 inches, as in this case, we get the following values of this ratio:

To compare this with our experiments, let us plot Table X. once more, rejecting, however, the end observations and completing the curve by the eye, thus getting rid of the error introduced at this point. We then find for this ratio, according to the different curves,

It is seen that these are all above the limit 2, as they should be, though it is possible that it may fall below in some cases owing to the variation of the permeability. As the magnetization increases, the values of the above ratio show that r decreases, as we should expect it to do from the variation of μ. To find the neutral point in this case, we must have in formula (10)

where x is the distance of the neutral point from the end. Making b-12, we have from this

By experiment we find that the neutral point is, in all the cases we have given in Table X, between 75 and 8.1 inches, which are quite near the points indicated by theory for the proper values of r, though we might expect curve D to pass through the point x=9, except for the disturbing causes we have all along considered.

Our formulæ, then, express the general facts of the distribution in this case with considerable accuracy.

These experiments and calculations show the change in distribution in an electro-magnet when we place a piece of iron

against one pole only. In an ordinary straight electro-magnet the neutral point is at the center. When a paramagnetic sub. stance is placed against or near one end, the neutral point moves toward it; but if the substance is diamagnetic it moves from it.

The same thing will happen, though in a less degree, in the case of a steel magnet, so that its neutral point depends on external conditions as well as on internal.

We now come to practically the most interesting case of distribution, namely, that of a straight bar magnetized longitudinally either by a helix around it, or by placing it in a magnetic field parallel to the lines of force; we shall also see that this is the case of a steel magnet magnetized permanently. This case is the one considered by Biot (Traité de Phys., tome iii. p. 77) and Green (Mathematical Papers of the late George Green, p. 111, or Maxwell's "Treatise," art. 439), though they apply their formulæ more particularly to the case of steel magnets. Biot obtained his formula from the analogy of the magnet to a Zamboni pile or a tourmaline electrified by heat. Green obtained his for the case of a very long rod placed in a magnetic field parallel to the lines of force, and, in obtaining it, used a series of mathematical approximations whose physical meaning it is almost impossible to follow. Prof. Maxwell has criticised his method in the following terms ("Treatise." art. 439):-"Though some of the steps of this investigation are not rigorous, it is probable that the result represents roughly the actual magnetization in this most important case." From the theory which I have given in the first part of this paper we can deduce the physical meaning of Green's approximation, and these are included in the hypotheses there given, seeing that when my formula is applied to the special case considered by Green, it agrees with it where the permeability of the material is great. My formula is, however, far more general than Green's.

It is to Green that we owe the important remark that the distribution in a steel magnet may be nearly represented by the same formula that applies to electro-magnets.

As Green uses what is known as the surface-density of mag. netization, let us first see how this quantity compares with those I have used.

Suppose that a long thin steel wire is so magnetized in the direction of its length that when broken up the pieces will have the same magnetic moment. While the rod is together, if we calculate its effect on exterior bodies, we shall see that the ends are the only portions which seem to act. Hence we may mathematically consider the whole action of the rod to be due to the distribution of an imaginary magnetic fluid over the ends

of the rod. As any case of magnetism can be represented by a proper combination of these rods, we see that all cases of this sort can be calculated on the supposition of there being two magnetic fluids distributed over the surfaces of the bodies, a unit quantity of which will repel another unit of like nature at a unit's distance with a unit of force. The surface-density at any point will then be the quantity of this fluid on a unitsurface at the given point, and the linear density along a rod will be the quantity along a unit of length, supposing the density the same as at the given point.

Where we use induced currents to measure magnetism we measure the number of lines of force, or rather induction, cut by the wire, and the natural unit used is the number of lines. of a unit-field which will pass through a unit-surface placed perpendicular to the lines of force. The unit-pole produces a unit-field at a unit's distance; hence the number of lines of force coming from the unit-pole is 47, and the linear density is

These really apply only to steel magnets; but as in the case of electro-magnets the action of the helix is very small compared with that of the iron, especially when it is very long and the iron soft,* we can apply these to the cases we consider. Transforming Green's into my notation, it gives

in which is Neumann's coefficient of magnetization by induction, and is equal to

Equation (5) can be approximately adapted to this case by making s∞, which is equivalent to neglecting those lines of force which pass out of the end section of the bar. This gives A'1, hence

* I take this occasion to correct an error in Jenkin's "Textbook of Electricity," where it is stated that, by the introduction of the iron bar into the helix, the number of lines of force is increased 32 times. The number should have been from a quite small number for a short thick bar and hard iron to nearly 6000 for a long thin bar and softest iron.

Now we have found (equation 7) that r=1

(15)

2

1

d

nearly,

πμ'

(16)

and this in Green's formula (equation 14) gives

which is identical with my own when u is large, as it always is in the case of iron, nickel, or cobalt at ordinary temperatures. When x is measured from the center of the bar, my equation becomes

H

λπ 4π/RR το

(17)

The constant part of Biot's formula is not the same as this; but for any given case it will give the same distribution.

Both Biot and Green have compared their formula with Coulomb's experiments, and found them to represent the distribution quite well. Hence it will not be necessary to consider the case of steel magnets very extensively, though I will give a few results for these farther on.

At present let us take the case of electro-magnets.

For observing the effect of the permeability, I took two wires 12.8 inches long and 19 inch in diameter, one being of ordinary iron and the other of Stub's steel of the same temper as when purchased. These were wound uniformly from end to end with one layer of quite fine wire, making 600 turns in that distance.

In finding A from Q", the latter was divided by 47AL, except at the end, where the end section was included with AL in the proper manner. x was measured from the end of the bar in inches.

The observations in Table XI. are the mean of four observations made on both ends of the bar and with the current in both directions.

The agreement with the formula in this table is quite good; but we still observe the excess of observation over the formula at the end, as we have done all along. Here, for the first time, we see the error introduced by the method of experiment which I have before referred to in the apparently small value of 47λ

at x=75.

On trying the steel bar, I came across a curious fact which, however, I have since found has been noticed by others. It is that when an iron or steel bar has been magnetized for a long time in one direction and is then demagnetized, it is easier to magnetize it again in the same direction than in the opposite direction. The rod which I used in this experiment had been used as a permanent magnet for about a month, but was demagnetized before use. From this rod five cases of distribution were observed: first, when the bar was used as an electro-magnet with the magnetization in same direction as the original magnetism; second, ditto with magnetization contrary to original magnetism; third, when used as a permanent magnet with magnetism the same as the original magnetism; fourth, ditto with magnetism opposite; and fifth, same as third but curve taken after several days. The permanent magnetism was given by the current.

The observations in Tables XI and XII. can be compared together, the quantities being expressed in the same unknown arbitrary unit. It is to be noted that the bars in Tables XI. and XII. were subjected to the same magnetizing-force.

First of all, from these tables and figures we notice the change in distribution due to the quality of the substance;