decreasing more slowly as the distances are greater, these ordinates will be bounded by a curve PQRTZ, which has its convexity turned toward the axis. We shall presently get full proof that this is the case here; but we premise this general view of the subject, that we may avoid the more tedious, but more philosophical, process of deducing the nature of the curve from the phenomena now under consideration. 45. This construction evidently makes the pair of ordinates Pp, Qq, equidistant with the pair Rr, Tt. Also, PP, Rr, and Q q, T t, are equidistant pairs. It is no less clear, that the sum of P p and T t, exceeds the sum of Q q and Rr. For if Cz be bisected in V, and V v be drawn perpendicular to it, cutting the straight lines PT and QR in x and y, then xv is the half sum of Pp and T t, and y v is the half sum of Qq and Rr. Moreover, if Qm and Tn are drawn parallel to the base, we see that Pm exceeds Rr; and, in general, that if any pair of equidistant ordinates are brought nearer to C, their difference increases, and vice versa. Also, if two pairs of equidistant ordinates be brought nearer to C, each pair by the same quantity, the difference of the nearest pair will increase more than the difference of the more remote pair. And this will hold true, although the first of the remote pair should stand between the two ordinates of the first pair. If the reader will take the trou ble of considering these simple consequences with a little attention, he will have a notion of all the effects that are to be expected in the mutual actions of the two bodies, sufficiently precise for our present purpose. We shall give a much more accurate account of these mathematical truths in treating the subject of MAGNETISM, where precision is absolutely necessary, and where it will be attended with the greatest success in the explanation of phenomena. 46. Now let us apply this to our present purpose. First, then, When the overcharged end of A is turned toward the undercharged end of B, A must be attracted; for Pp+Tt is greater than Qq + Rr. 47. Secondly. This attraction must increase by bringing the bodies nearer; for this will increase the difference between Pm and Rn. 48. Thirdly. The attraction will increase by increasing the length either of A or of B (the distance Ns remaining the same); for by increasing the length of A, which is represented by pr or qt, Rr is more diminished than Tt is. In like manner, by increasing B, whose length is represented by pq or rt, we diminish Qq more than T t. 49. On the other hand, if the overcharged end of B front the overcharged end of A, their mutual action will be F'f'(-Pp Qq Rr — Tt) Ff - and A will be repelled, and the repulsion will increase or diminish, by change of distance or magnitude, precisely in the same manner that the attractions did. It is hardly necessary to observe, that all these consequences will result equally from bringing an apparatus similar to that represented in fig. 3. near to another of the same kind; and that they will be various according to the position and the redundancy or deficiency of the two parts of each apparatus. 50. If the body B of fig. 5, is not at liberty to approach toward A, nor to recede from it, and can only turn round its centre B, it will arrange itself in a certain determinate position with respect to that of A. For example, if the centre B (fig. 7.) be placed in the line passing through S and N of the body A, B will arrange itself in the same straight line for if we forcibly give it another position, such as s Bn, N will attract s and repel n, and these actions will concur in putting B into the position s' B n'. S, however, will repel s and attract n; and these forces tend to give the contrary position. But S being more remote than N, the former forces will prevail, and B will take the position s' Bn'. : If the centre B be placed somewhere on the line AD, drawn through a certain point of the body NAS, (which will be determined afterwards), at right angles to NAS, the body B will assume the position n' Bs', parallel to NAS, but subcontrary. For if we forcibly give it any other position n Bs, it is plain that N repels n and attracts s, while S attracts n and repels s. These four forces evidently combine to turn the body round its centre, and cannot balance each other till B assume the position n' B s', where n' is next to S, and s' is next to N. If the centre of B have any other situation, such as B', the body will arrange itself in some such position as n' B's', It may be demonstrated, that if B be infinitely small, so that the action of the end of A on each of its extremities may be considered as equal, B will arrange itself in the tangent BT of a curve NB'S, such that if we draw NB', SB', and from any point T of the tangent draw TE parallel to B'N, and TF parallel to B'S, we shall have B'E to B'F, as the force of S to the force of N. This arrangement of B will be still more remarkable and distinct if N be an overcharged sphere, and S an undercharged one, and both be insulated. We must leave it to the reader's reflection to see the changes which will arise from the inequality of the redundancy and deficiency in A or B, or both, and proceed to consider the consequences of the mobility of the electric fluid. These will remove all the difficulty and paradox that appears in some of the foregoing propositions. 51. Let the body A (fig. 4.) contain redundant fluid, and let B be in its natural state, but let the fluid in A be fixed, and that in B perfectly moveable; it is evident that the redundant fluid in A will repel the moveable fluid in B, toward its remote extremity N, and leave it undercharged in S. The fluid will be rarefied in S, and constipated in N. We need only consider the mutual actions of the redundant fluid and redundant matter. It is plain that things are now in the situation described in § 15: A must be attracted by B, because f'm', and ≈ is greater than 2'. The attractive force is F'f' x ( — %'). Thus we see that the hypothesis is accommodated to the phenomena in the case in which it appeared to differ so widely from it. Had the fluid been immoveable, the mutual actions would have so balanced each other that no external effects would have appeared. But now the greater vicinity of the redundant matter prevails, A is attracted by B, and, the actions being all mutual, B is attracted by A, and approaches it. 52. We have supposed that the fluid in A is immoveable; but this was for the sake of greater simplicity. Suppose it moveable. Then, as soon as the uniform distribution of the fluid in B is changed, and B becomes undercharged at s, and overcharged at n, there are forces acting on the fluid in A, and tending to change its state of distribution. The redundant matter in S attracts the redundant fluid in A more than the more remote redundant fluid in n repels it, because' is less than . This tends to constipate the redundant fluid of A in the nearer parts, and render N more redundant, and S less redundant in fluid than before. It is plain, that this must increase their mutual action, without changing its nature. It can be strictly demonstrated, that however small the redundancy in A may be, it can never be rendered deficient in its remote extremity by the action of the unequally disposed fluid in B, if the fluid in B be no more nor less than its natural quantity. It is also plain, that this change in the disposition of the fluid in A must increase the similar change in B. It will be still more rarefied in s, and condensed in n; and this will go on in both till all is in equilibrio. When things are in this state, a particle of fluid in B is in equilibrio by the combined action of several forces. The particle B is propelled toward n by the action of the redundant fluid in A. But it is urged toward S by the repulsion of the redundant fluid on the side of n, and also by the attraction of the redundant matter on the side of s; and the repulsion of the redundant fluid in A must be conceived as balancing the united action of those two forces residing in B. 53. Hence we may conclude, that the density of the fluid in B will increase gradually from s to n. It will be extremely difficult to obtain any more precise idea of its density in the different parts of B, even although we knew the law of action between single particles. n This must depend very much on the form and dimensions of B; for any individual particle sustains the sensible action of all the redundant fluid and redundant matter in it, since we suppose it affected by the more remote fluid in A. All that we can say of it in general is, that the density in the vicinity of s is less than the natural density; but in the vicinity of n it is greater; and therefore there must be some point between s and ʼn where the fluid will have its natural density. This point may be called a NEUTRAL point. We do not mean by this that a particle of superficial fluid will neither be attracted nor repelled in this place. This will not always be the case (although it will never be greatly otherwise); nor will the variation of the density in the dif ferent parts of B be proportional to the force of A on those parts. Some eminent naturalists have been of this opinion; and, having made experiments in which it appeared to be otherwise, they have rejected the whole theory. But a little reflection will convince the mathematician, that the sum of the internal forces which tend to urge a particle of fluid from its place, and which are balanced by the action of A, are not proportional to the variations of density, although they increase and decrease together. We shall take the proper opportunity of explaining those experiments; and will also consider some simple, but important cases, where we think the law of distribution of the fluid ascertained with tolerable precision. If we suppose, on the other hand, that A is undercharged, the redundant matter in A will attract the moveable fluid in |