and this is indeed what we ought to expect, from the perfect analogy which we have remarked from the beginning between magnets and poles of this kind. The formulas deduced from this approximation enable us to trace the variations of the magnetic charge in wires of the same magnitude, but of unequal lengths, and in wires of the same length but of unequal magnitude; and, in short, in wires of any magnitude and length whatever, by supposing them always magnetized by the method of double touch. The conclusion derived from a comparison of the calculated and observed results cannot, however, be explained here; but the reader will find it in my Traité de Physique.

212. The experiments of Coulomb, upon which these calculations are founded, present an equal and opposite distribution of magnetism in the two halves of the needle; a distribution which is indeed the most advantageous for obtaining a considerable directive force, and which, therefore, we should endeavour as much as possible to effect. Experience, however, informs us that this is impossible in tempered needles, when their length is very great, compared with the diameter of their transverse section.

In this case, whatever method of magnetizing is employed, several centres are formed, the developement of which is probably owing to the reaction of the poles upon the points near the centre. In this case, the curve of intensity is no longer situated for the two halves of the needle on different sides of theaxis. It necessarily undulates above and below, as represented in figure 120; and, consequently, its form can no longer be represented by the same analytical expression as before. Fortunately, there is every reason to believe that this limitation is not to be regretted. For, in the first place, it does not happen in annealed needles, unless, perhaps, they very much exceed in length those which are ordinarily used; and with respect to tempered needles, if we are not constrained by some urgent motive to make them extremely light, there will always be an advantage in giving them a sufficient thickness, in order that the free magnetism may be of the same nature in each of their halves; for, with an equality of coercive force, the developement of new centres always weakens the statical moment of the directive force for each half of the needle, and renders the action less energetic at equal distances from the poles.

It is obvious, in general, that the distribution of magnetism in a needle, and the absolute degree of saturation of which it is susceptible, depend not only on its dimensions, but also on the higher or lower temper which it has received. Coulomb had studied the influence of this last circumstance. He shows that we must always begin by tempering the needle at a white heat, whatever be its dimensions, and then, if its length is less than thirty times its thickness, we must leave it at this temper; but if it exceeds this proportion, we must bring it back again, by annealing it to the state of dark red, in order to avoid a multiplication of centres which its great length might occasion.

Of the best Forms of Compass Needles.

213. THE results at which we have arrived in the preceding sections, should serve to direct us in making needles for compasses. Although this application may be very easy, its importance entitles it to particular attention; and this will be the more readily bestowed, as here also Coulomb is our guide.

The compasses commonly used, whether designed for land or sea, are formed of needles artificially magnetized, and provided with a cap at the centre, which rests on a pivot of some metal not magnetic. A little counterpoise placed on one arm of the needle renders it horizontal. It is necessary to change the place or size of this counterpoise as we change our latitude, the moment of the vertical forces of terrestrial magnetism being different in different latitudes. Whatever be the form of the needle, it is easy to determine, on its surface, the horizontal direction of the magnetic resultant by the method already explained. If the needle move on its pivot with perfect freedom, it will naturally direct itself in such a manner as to cause the magnetic axis to correspond exactly to the magnetic meridian; and consequently, when once known, it will exactly determine this meridian. But the friction of the pivot on the bottom of the cap opposes this tendency, and presents an obstacle which the directive force must surmount in order to bring the needle to the magnetic meridian; whence it is evident, that the best construction is that in which the friction is least, and the directive force the greatest. E. & M.

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214. On the supposition that the pivots and caps are of the same shape, the same materials, and formed with equal care, the friction will depend simply on the weight of the needle; and it may be measured by presenting the needle from a distance, while balanced on its pivot, to a magnet that draws it from the plane of the magnetic meridian, and observing how nearly it returns to its proper situation, when left at perfect liberty. It should seem that the arcs which it describes on each side of this plane, a great number of experiments being used, should be proportional to the force of friction. By observations of this kind Coulomb found that for very sharp pivots, and caps formed of a substance sufficiently hard, the friction is proportional to the power of the pressure.

But when by long use the pivots have become blunted, and as it were fitted to the excavation of the cap, which is frequently the case, he found the friction to be simply proportional to the pressure. This is the first established fact of which we are to avail ourselves. Let us conceive a magnetic needle of any form and size whatever, 'placed on a pivot of the above description; and, without changing its length at all, let us only double its thickness, or which amounts to the same thing, cover it with another lamina of metal precisely similar; the pressure on the pivot will be doubled, and also the friction; but not the directive force. For it is manifest, and proved by experiment, that this force increases in a less ratio than the thickness, since the re-action of homologous poles on each other destroys a part of the free magnetism which each one separately possessed. The needle, when covered with its additional coating, will point out the magnetic meridian less accurately than before; and hence it will be seen that, other things being the same, the most correct needles are those of the least diameters. The diameter will be sufficiently great if it be such as to prevent the needle from being bent by its weight.

215. Let us now proceed to consider the lengths of needles, and first, the case of those which from their dimensions and physical state possess only one kind of free magnetism in each of their ends. Then the analytical law relative to the intensities, obtained above, shows that unless the needles are exceedingly short, their directive forces, the diameters being equal, are proportional to their lengths, at least if we suppose their transverse

sections to be every where the same. But, in this case, the weight, and consequently the friction which results from it, are each proportional to the length. So that if we avoid exceedingly small dimensions, all needles, whatever be their lengths, have nearly the same degree of accuracy. This, however, is true. only on the supposition of a symmetrical distribution of magnetism in the two arms of the needles, and a freedom from consecutive points. It is necessary then to attend to the relation of the length to the thickness, as well as to the state of annealing and tempering, in order that this condition may be fulfilled; and we must accordingly observe the directions given in the preceding section. If the length of the needle be less than 30 times its thickness, we must temper it at a white heat, before we develope its magnetic power. If, on the other hand, its length exceed this proportion, we must anneal it till it becomes of a dark red colour. When the length is between these two limits, it is not of much consequence which process we employ. The superiority possessed by needles having a single magnetic centre over those of several centres is incontestible, if we suppose the same quantity of magnetism to be developed on the whole in each case. But it is not impossible, that with other proportions of thickness and length, and other degrees of temper, the diminution of magnetic force occasioned by the multiplicity of centres, may be compensated by the existence of a coercive force more considerable than could be otherwise obtained, or by a more abundant developement of magnetism. It appears that Coulomb performed a great many experiments relative to this subject, which he proposed to arrange in tables, so that we might know beforehand what were the most favourable circumstances for every variety of dimensions in the needles. But unfortunately, nothing has been found in his manuscripts sufficiently matured to be employed in so important an undertaking; and the subject still demands the attention of philosophers.

216. It now remains for us to inquire, which is the most ad vantageous of all the forms of needles. Usually, they are par. allelograms, cylinders, or arrows. Coulomb ascertained by experiment, that when the weights are equal, arrow-shaped needles have the greatest directive force. And this might be naturally inferred from the reason which induced him to arrange his magnetic bundles by steps retreating in the direction of their

thickness, as already explained. It will also be evident from the same principles, that there is great disadvantage arising from the extremities of the needles being enlarged; and this modification, which some have proposed to introduce, should be steadily opposed. The remarks here made are equally applicable to dipping needles. With respect to these it will also be necessary to employ the processes for correction by inversion, as heretofore made known.

Of the Action of Magnets on other Natural Substances.

217. WE have said that iron, steel, nickel, and cobalt, were the only magnetic metals at present known. And indeed they are the only metals capable of acquiring a high and permanent degree of magnetism. Still if we take a small needle a third of an inch in length, and of about an inch in thickness, of any substance whatever, and suspend it by a silk thread between the opposite poles of two powerful magnets, as represented in figure 121, it will be found always to place itself in the direction of these poles; and if we cause it to vibrate about its line of equilibrium, the oscillations performed in presence of the magnets are much more rapid than those which take place in empty space. These little needles then are sensible to the influence of magnetism. We shall be equally successful whether we employ in our experiments needles of gold, silver, glass, wood, or any other substance, organic or inorganic. These remarkable facts were discovered by Coulomb, and announced by him to the National Institute in May, 1812.†

218. There seem to be only two ways of accounting for these phenomena. Either all substances in nature are susceptible of magnetism, or all possess particles of iron, or some other magnetic metal from which this property is derived. But this alternative is not so necessary as we should at first suppose; for it rests on the assumption, that the action exhibited by the needles

† A detailed account of them is given in Biot's Traité de Physique, with a calculation of the forces exerted by them.