will be completely determined, by satisfying the equation (i) and the condition (c). Let us then assume DF (y C0s vr — xainvr; z) = hD<f> (y cos «r — * sin w); h being a quantity independent of x, y, z, and aee if it be possible to determine A so as to satisfy the condition (c). Now on this supposition D V— UV— hD<f> (y cos «r — x sin w)—(x sin $—y cos The value of Dp' corresponding to this potential function is Dp m 0, and on account of the parallelism of the lines L, IS, dec. to each other, and to A'b equator da — d»rx. The condition (c) thus becomes Q=Dp + DPt («'): Dp and Dpt being the elementary densities on A'a surface at opposite ends of any of the lines L, L', &c. corresponding to the potential function DV—D'V. But it is easy to see from the form of this function, that these elementary densities at opposite ends of any line perpendicular to a plane whose equation is 0 = y (h cos v + b cos — x (h sin vr + b sin <f>), are equal and of contrary signs, and therefore the condition (c) will be satisfied by making this plane coincide with that perpendicular to L, L', &c., whose equation, as before remarked, is O = a;cosw + ysinw; that is the condition (c) will be satisfied, if A be determined by the equation h cos «r + b cos <f> h sin vr + b sin <f> which by reduction becomes 0 = h + b cos ((f> - w), and consequently V+1)V= ft (xcoav+y sinw) + hD<f> {y casta — x sin w) »= 0x jcos vr + ^ sin «. cos — w) Ztyj + £y jsin tsr — j| cos «. cos (</, — w) Ztyj, ■» fix COt jw — ^ cos — «r) D^j + sin j*r — ^ cos - m) Ztyj. When therefore <f> is augmented by the infinitely small angle D<f>, w receives the corresponding increment — ^ cos (<f> — w) Z><£, and the form of V remains unaltered; the preceding reasoning is consequently applicable to every instant, and the general relation between <f> and « expressed by 0 = Bv + ~cos (<f>- v) D<f>: a common differential equation, which by integration gives sin (£tt + £7 + i«r -' 5" being an arbitrary constant, and 7, as in the former part of this article, the smallest root of 0 = b sin 7 — /9. Let tsr0 and <f>, be the initial values of w and ^; then the total potential function at the next instant, if the electric fluid remained fixed, would be V,~ft{x cos »0 + y sin «r0) + (a; sin — 7 cos W^, and the whole force to move a particle j>, whose co-ordinates are x, y, z, which, in order that our solution may he applicable, must not be less than /S, and consequently the angle <f>t — vra must be between 0 and ir: when this is the case, w is immediately determined from 0 by what has preceded. In fact, by finding the value of H from the initial values and <f>t, and making £= iir + i7 + £*r — l<f>, we obtain . „ tan 6~e(*-*,>«>'Y + tan7tan^{e^-*,>coti'- 1}' £ being the initial value of £ We have, in the latter part of this article, considered the body A at rest, and the line X', parallel to the direction of b, as revolving round it: but if, as in the former, we now suppose this line immovable and the body to turn the contrary way, so that the relative motion of X' to X may remain unaltered, the electric state of the body referred to the axes X, Y, Z, evidently depending on this relative motion only, will consequently remain the same as before. In order to determine it on the supposition just made, let X' be the axis of x, one of the co-ordinates of p, referred to the rectangular axes X', Y, Z, also y, z, the other two; the direction X! Y, being that in which A revolves. Then, if v be the angle the system of lines L, L', &c. forms with the plane (as', *), we shall have «r + «.' = <f>; <f>, as before stated, being the angle included by the axes X, X'. Moreover the general values of V and f will be F=/9(a;'cos«r' + ysinw') and £= Jrr + J 7 — \ia, and the initial condition, in order that our solution may be applicable, will evidently become $0 — «r, = «.'„ = a quantity betwixt 0 and 7r. As an example, let tan 7 = ~, since we know by experiment that 7 is generally very small; then taking the most unfavourable case, viz. where w'a = 0, and supposing the body to make one revolution only, the value of f, determined from its initial one, = \tt + $7 - |w'0, will be found extremely small and only equal to a unit in the 27tfe decimal place. We thus see with what rapidity f decreases, and consequently, the body approaches to a permanent state, defined by the equation O-C-Jr + fr-Jw'. Hence, the polarity induced by the rotation is ultimately directed along a line, makiug an angle equal to fair + y with tbe axis X which agrees with what was shown in the former part of this article. The value of V at the body's surface being thus known at any instant whatever, that of the potential function at a point p exterior to the body, together with the forces acting there, will be immediately determined as before. APPLICATION OF THE PRELIMINARY RESULTS TO THE THEORY OF MAGNETISM. f 14.) The electric fluid appears to pass freely from one part of a continuous conductor to another, but this is by no means the case with the magnetic fluid, even with respect to those bodies which, from their instantly returning to a natural state the moment the forces inducing a magnetic one are removed, must be considered, m a certain sense, as perfect conductors of magnetism. Coulomb, I believe, wa8 the first who proposed to consider these as formed of an infinite number of particles, each of which conducts the magnetic fluid in its interior with perfect freedom, but which are so constituted that it is impossible there shall be any communication of it from one particle to the next. This hypothesis is now generally adopted by philosophers, and its consequences, as far as they have hitherto been developed, are found to agree with observation; we will therefore admit it in what follows, and endeavour thence to deduce, mathematically, the laws of the distribution of magnetism in bodies of any shape whatever. Firstly, let us endeavour to determine the value of the potential function, arising from the magnetic state induced iu a very small body A, by the action of constant forces directed parallel to a given right line; the body being composed of an infinite. number of particles, all perfect conductors of magnetism and' originally in a natural state. In order to deduce this more immediately from Art. 6S we will conceive these forces to arise |