if there were any substances in nature whose magnetic powers, like those of iron and nickel, admit of considerable developeinent, and in which moreover the coercive force was, as we have here supposed it, the same for all their elements, the results of the preceding theory ought scarcely to differ from what would be observed in bodies formed of such substances, provided no one of their dimensions was very small, compared with the others. The hypothesis of a constant coercive force was adopted in this article, in order to simplify the calculations: probably, however, this is not exactly the case of nature, for a bar of the hardest steel has been shown (I think by Mr Barlow) to have a very considerable degree of magnetism induced in it by the earth's action, which appears to indicate, that although the coercive force of some of its particles is very great, there are others in which it is so small as not to be able to resist the feeble action of the earth. Nevertheless, when iron bodies are turned slowly round their axes, it would seem that our theory ought not to differ greatly from observation; and in particular, it is very probable the angle y might be rendered sensible to experiment, by sufficiently reducing b the component of tlie force/. The remaining articles treat of the theory of magnetism. This theory is here founded on an hypothesis relative to the constitution of magnetic bodies, first proposed by Coulomb, and afterwards generally received by philosophers, in which they are considered as formed of an infinite number of conducting elements, separated by intervals absolutely impervious to the magnetic fluid, and by means of the general results contained in the former part of the Essay, we readily obtain the necessary equations for determining the magnetic state induced in a body of any form, by the action of exterior magnetic forces. These equations accord with those M. Poissox has found by a very different method. (Mem. de l'Acad. des Sciences, 1821 et 1822.) If the body in question be a hollow spherical shell of constant thickness, the analysis used by Laplace (m^c. Ce"l. Liv. 3) is applicable, and the problem capable of a complete solution, whatever may be the situation of the centres of the magnetic forces acting upon it. After having given the general solution, we have supposed the radius of the shell to become infinite, its thickness remaining unchanged, and have thence deduced formula; belonging to an indefinitely extended plate of uniform thickness. From these it follows, that when the point p, and the centres of the magnetic forces are situate on opposite sides of a soft iron plate of great extent, the total action on p will have the same direction as the resultant of all the forces, which would be exerted on the points p, p, p",p", etc. in infinitum if no plate were interposed, and will be equal to this resultant multiplied by a very small constant, quantity: the points p, p, p", p"\ &c. being all on a right line perpendicular to the flat surfaces of the plate, and receding from it so, that the distance between any two consecutive points may be equal to twice the plate's thickness. What has just been advanced will be sensibly correct, on the supposition of the distances between the point p and the magnetic centres not being very great, compared with the plate's thickness, for, when these distances are exceedingly great, the interposition of the plate will make no sensible alteration in the force with which p is solicited. When an elongated body, as a steel wire for instance, has, under the influence of powerful magnets, received a greater degree of magnetism than it can retain alone, and is afterwards left to itself, it is said to be magnetized to saturation. Now if in this state we consider any one of its conducting elements, the force with which a particle p of magnetism situate within the element tends to move, will evidently be precisely equal to its coercive force f, and in equilibrium with it. Supposing therefore this force to be the same for every element, it is clear that the degree of magnetism retained by the wire in a state of saturation, is, on account of its elongated form, exactly the same as would be induced by the action of a constant force, equal to/, directed along lines parallel to its axis, if all the elements were perfect conductors; and consequently, may readily be determined by the general theory. The number and accuracy of Coulomb's experiments on cylindric wires magnetized to saturation, rendered an application of theory to this particular case very desirable, in order to compare it with experience. We have therefore effected this in the last article, and the result of the comparison is of the most satisfactory kind. GENERAL PRELIMINARY RESULTS. (1.) The function which represents the sum of all the electric particles acting on a given point divided by their respective distances from this point, has the property of giving, in a very simple form, the forces by which it is solicited, arising from the whole electrified mass.—We shall, in what follows, endeavour to discover seme relations between this function, and the density of the electricity in the mass or masses producing it, and apply the relations thus obtained, to the theory of electricity. Firstly, let us consider a body of any form whatever, through which the electricity is distributed according to any given law, and fixed there, and let x, y, z, be the rectangular co-ordinates of a particle of this body, p the density of the electricity in this particle, so that dx'dy'dz being the volume of the particle, p dx'dy'dz' shall be the quantity of electricity it contains: moreover, let r be the distance between this particle and a point p exterior to the body, and V represent the sum of all the particles of electricity divided by their respective distances Irom this point, whose co-ordinates are supposed to be x, y, z, then shall we have r' = V(*' - *)*+ (yr- »• + (7 -1)\ and) >j'j.jjJ the integral comprehending every particle in the electrified mass under consideration. Laplace has shown, in his Mec. Celeste, that the function V has the property of satisfying the equation a_ePV <PV d*Y and as this equation will be incessantly recurring in what fellows, we shall write it in the abridged form 0 = $V; the symbol S being used in no other sense throughout the whole of this Essay. In order to prove that 0—SV, we have only to remark, that by differentiation we immediately obtain 0 = S -, and consequently each element of F substituted for V in the above equation satisfies it; hence the whole integral (being considered as the sum of all these elements) will also satisfy it. This reasoning ceases to hold good when the point p is within the body, for then, the coefficients of some of the elements which enter into V becoming infinite, it does not therefore necessarily follow that. V satisfies the equation 0 = oT, although each of its elements, considered separately, may do so. In order to determine what 8 V becomes for any point within the body, conceive an exceedingly small sphere whose radius is a inclosing the point p at the distance b from its centre, a and b being exceedingly small quantities. Then, the value of V may be considered as composed of two parts, one due to the sphere itself, the other due to the whole mass exterior to it: but the last part evidently becomes equal to zero when substituted for V in SV, we have therefore only to determine the value of 8 V for the small sphere itself, which value is known to be p being equal to the density within the sphere and consequently to the value of p at p. If now xlt y„ z,, be the co-ordinates of the centre of the sphere, we have and consequently B (2ira*p — f 7rJ*p) =» — iirp. Hence, throughout the interior of the mass 0 = BV+4>>rp; of which, the equation 0 = 87" for any point exterior to the body is a particular case, seeing that, here p — 0. Let now q be any line terminating in the point p, supposed without the body, then — (-j- J = the force tending to impel a particle of positive electricity in the direction of q, and tending to increase it. This is evident, because each of the elements of V substituted for V in — (-t-j , will give the force arising from this element in the direction tending to increase q, and consequently, — (-7—) will give the sum of all the forces due to every element of V, or the total force acting on p in the same direction. In order to show that this will still hold good, although the point p be within the body; conceive the value of V to be divided into two parts as before, and moreover let p be at the surface of the small sphere or b = a, then the force exerted by this small sphere will be expressed by fda\ da being the increment of the radius a, corresponding to the increment dq of q, which force evidently vanishes when a « 0: we need therefore have regard only to the part due to the mass exterior to the sphere, and this is evidently equal to But as the first differentials of this quantity are the same as those of Fwhen a is made to vanish, it is clear, that whether the point p be within or without the mass, the force acting upon it in the direction of q increasing, is always given by — (-arj • |