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 Eth 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 bis piece of soft iron F varied as some unknown power n of the surface-density o. Then we find F=Cɛari, 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 courbes 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 1. rd. in a rod of circular section. Now the quantity p="is nearly constant in these rods for the same quality of steel, whencer 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 =wRRET, . . . . (18) 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. Jarnin 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. 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 ve H 1 11- (19) 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 Marc {z-chip cons, . (20) where C=*=PVT This only attains a maximum when =00, or the rod is infinitely 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 6 C6_1 0o=yn2R'pchili: : : (21) which attains a maximum as before when = 0. When the attracted body is large, the attraction will depend more nearly upon the linear density &C621 1=ac, Rui -, . . . (22) which is a maximum when ==- 6 142 For the third case we have the value of Q" at the center of the bar from equation (6), 0 H (E4C62 - 1)2 . . . (23) The condition for a maximum gives in this case 6 1.85 7. For the last case in which the magnetic moment for a given length is to be made a maximum, we find 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 15 . . . . (24) 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 rel. ative 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 indi. cated 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 furmula, but would be found so for the exact one. Thus for the “normal magnet” the distribution becomes 1=C(31-8-31), 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.) H. A. Rowland-Studies on Jagnetic Distribution. 107 **** This distribution is not the same as that given by M Jamin; but as bis method is so defective, and his “normal magnet" s) indetinite, the agreement is sufficiently near. The surface-density at any point of a magnet is oo, . 872pR' Firepi . (25) which, for the same kind of steel, is dependent only on and Hence in two similar magnets the surface-density is the same at similar points, the linear density is proportional to the linear dimensions, the surface integral of magnetic induction over half the magnet or across the section is proportional to the surface dimensions of the magnets, and the magnetic moments to the voluines of the magnets. The forces at similar points with regard to the two magnets will then be the same. All these remarks apply to soft iron under induction providing the inducing force is the same, and hence include Sir William Thomson's well-known law with regard to similar electromagnets; and they are accurately trur notwithstanding the approximate nature of the formula from which they have here been deduceil. Our theory gives us the means of letermining what effect the boring of a hole through the center of a magnei would have. In this case R' is not much affected, but R is increased. Where the magnet is used merely to affect a compass-needle, we should then see that the hole through the center has little effect where the magnet is short and thick; but where it is long, the attraction on the compass-needle is much diminished. Where the magnet is of the U-form, and is to be used for sustaining weights, the practice is detrimental, and the sustaining power is diminished ** ** Le man helt |