above by the freer contrary wind (assuming as before a surface retardation), the sound-beams are curved upward, and the lowest ray that can reach the distance of the observer at o, is that which touching the surface of the sea is gradually so tilted up

W

E

2.-Adverse Wind.

ward that it passes above the ear of the listener, leaving him practically in an acoustic shadow; very much as an observer on the deck of a vessel when losing the sight of the hull of another vessel ten miles off, by reason of the interposed convexity of the ocean, stands in the optical shadow of the earth. In both cases if the conditions favor, the boundary of the shadow may be re-crossed by ascending from the deck to the mast-head, and the sight or the sound-beam thus regained.

W

E

3.-Compound Wind.

Fig. 3 represents the disturbing effect of a lower contrary wind with an opposite wind above. In this case the principal result will be a depression of the sound-beam as in fig. 1, but more strongly marked, as the differences of motion as we ascend will be more rapid. Attending this action, however, there will probably be some lagging of the lower stratum of the adverse wind by reason of the surface friction; the tendency of which will be to slightly distort the lower sound radiations, by giving them a reverse or serpentine curvature. The upper rays of sound would probably have only a single declining curvature, similar to that shown in fig. 1.

One result of this condition of the locus of the normals (to use a mathematical phrase) would be to make the sound less andible (or possibly sometimes inaudible) at a point (as at t) midway between the two stations. This hypothetical case of compound refraction would appear to offer a plausible explanation, not only of the paradox of a nearer trumpet-sound being diminished in power by the wind which increased the effect of a more distant whistle, but also of the puzzling "belt" of inaudibility previously noticed. Duane [D], and Henry [8].

Numerous other cases might be represented by diagrams, as of a sound being hindered or tilted upward by a concurrent wind of unequal velocity, or downward by an opposing wind of similar character, and of the various permutations of differing currents in oblique directions; to which might be added various resultants of unequal motion producing lateral refraction, but this is unnecessary. Enough has been said, it is hoped, to clear from popular misapprehension, the admirable hypothesis of Professor Stokes, raised by the equally admirable investigations of Professor Henry, to the rank of a "theory;" and to show that it has a real and demonstrated basis, or in other words that it is a vera causa. The question of its sufficiency lies entirely within the grasp of mathematical discussion; but a long series of accurate and comprehensive observations will yet be required to discover its full compass of practical result, and to determine its precise limit of capacity in subjugating the "abnormal phenomena" of sound.

ART. VIII-Studies on Magnetic Distribution; by HENRY A. ROWLAND, of the Johns Hopkins University, Baltimore.

(Continued from page 29.)

VI.

M. JAMIN, in his recent experiments on magnetic distribution, has obtained some very interesting results, although I have shown his method to be very defective. In his experiments on iron bars magnetized at one end, he finds the formula to apply to long ones as I have done. Now it might be argued that as the two methods apparently give the same result, they must be equally correct. But let us assume that the attraction of his piece of soft iron F varied as some unknown power n of the surface-density d. Then we find

F=C&L

which shows that the attractive force or any power of that force can be represented by a logarithmic curve, though not by the same one. Hence the error introduced by M. Jamin's method is insidious and not easily detected, though it is none the less hurtful and misleading, but rather the more so.

However, his results with respect to what he calls the normal magnet* are to some extent independent of these errors; and we may now consider them.

Thus, in explaining the effect of placing hardened steel

* "On the Theory of Normal magnets," Comptes Rendus, March 31, 1873, translated in Phil. Mag., June, 1873.

plates on one another, he says: "Quand on superpose deux lames aimantées pareilles, les courbes qui représentent les valeurs de F (the attractive force on the piece of soft iron) s'elévent, parce que le magnétisme quitte les faces que l'on met en contact pour réfugier sur les parties extérieures. En même temps, les deux courbes se rapprochent l'une de l'autre et du milieu de l'aimant. Cet effet augmente avec une troisième lame et avec une quatriéme. Finalement les deux cour

bes se joignent en milieu."

In applying the formula to this case of a compound magnet, we have only to remark that when the bars lie closely together, they are theoretically the same as a solid magnet of the same section, but are practically found to be stronger, because thin bars can be tempered more uniformly hard than thick ones. The addition of the bars to each other is similar to an increase in the area of the rod, and should produce nearly the same effect on a rod of rectangular section as the increase of diameter in a rod of circular section. Now the quantity p= is nearly

rd.

2

constant in these rods for the same quality of steel, whence r decreases as d increases; and this in equation (17) shows that as the diameter is increased, the length being constant, the curves become less and less steep until they finally become straight lines. This is exactly the meaning of M. Jamin's remark.

Where the ratio of the diameter to the length is small, the curves of distribution are apparently separated from each other, and are given by the equation

which is not dependent on the length of the rod. This is exactly the result found by Coulomb (Biot's Physique, vol. iii, pp. 74, 75.) M. Jamin has also remarked this. As he increases the number of plates, he states that the curves approach each other and finally unite; this he calls the "normal magnet;" and he supposes it to be the magnet of greatest power in proportion to its weight. "From this moment," says he, "the combination is at its maximum." The normal magnet as thus defined is very indefinite, as M. Jamin himself admits.

By our equations we can find the condition for a maximum, and can give the greatest values to the following, supposing the weight of the bar to be a fixed quantity in the first three. 1st. The magnetic moment.

2d. The attractive force at the end.

3d. The total number of lines of magnetic force passing from the bar.

4th. The magnetic moment, the length being constant and the diameter variable.

Either of these may be considered as a measure of the power of the bar according to the view we take. The magnetic moment of a bar is easily found to be

M=

5 4πr2R' 2

{

and if y is the weight of a unit of volume of the steel and W is the weight of the magnet, we have finally

nitely long compared with its diameter. The second case is rather indefinite, seeing that it will depend upon whether the body attracted is large or small. When it is small, we require to make the surface-density a maximum, the weight being constant.

We find

d

=∞. When the

attracted body is large, the attraction will depend more nearly upon the linear density

3

λο

(22)

For the third case we have the bar from equation (6),

the value of Q" at the center of

p

(23)

For the last case in which the magnetic moment for a given length is to be made a maximum, we find

b •1
P

This last result is useful in preparing magnets for determining the intensity of the earth's magnetism, and shows that the magnets should be made short, thick, and hard for the best effect.*

But for all ordinary purposes the results for the second and third cases seem most important, and lead to nearly the same result; and taking the mean we find for the maximum magnet

We see from all our results that the ratio of the length of a magnet to its diameter should vary inversely as the constant p. This constant increases with the hardness of the steel, and hence the harder the steel the shorter we can make our mag. nets. It would seem from this that the temper of a steel magnet should not be drawn at all, but the hardest steel used, or at least that in which p was greatest. The only disadvan tage in using very hard steel seems to be the difficulty in imparting the magnetism at first, and this may have led to the practice of drawing the temper; but now when we have such powerful electromagnets, it seems as if magnets might be made shorter, thicker and barder, than is the custom. With the relative dimensions of magnets now used, however, hardening might be of little value.

We can also see from all these facts, that if we make a compound magnet of hardened steel plates there will be an advantage in placing more of them together, thus making a thicker magnet than when they are softer. We also observe that as we pile them up the distribution changes in just the way indicated by M. Jamin, the curve becoming less and less steep.

Substituting in the formula the value of p which we have found for Stub's steel not hardened, but still so hard as to rapidly dull a file, we find the best ratio of length to diameter to be 33.8, and for the same steel hardened about 17, though this last is only a rough approximation. This gives what M. Jamin has called the normal magnet. The ratio should be less for a U-magnet than for a straight one.

For all magnets of the same kind of steel in which the ratio of length to diameter is constant the relative distribution is the same; and this is not only true for our approximate formula, but would be found so for the exact one.

Thus for the "normal magnet" the distribution becomes

where C is a constant, and x is measured from the center. The distribution will then be as follows:

*Weber recommends square bars eight times as long as they are broad, and tempered very hard. (Taylor's Scientific Memoirs, vol. ii, p. 86.)