The value of the total potential function at any point p within the shell, whose polar co-ordinates are r, e, w, is Απ 39 (1 − g) (6) = (1 − g) (1 + 2g) ΣU ® (2i + 1)3 još In a similar way, the value of the same function at a point p' exterior to the shell, all the inducing bodies being within it, is found to be r, and w in this expression representing the polar co-ordinates of p'. To give a very simple example of the use of the first of these formulæ, suppose it were required to determine the total action exerted in the interior of a hollow spherical shell, by the magnetic influence of the earth; then making the axis of x to coincide with the direction of the dipping needle, and designating by f, the constant force tending to impel a particle of positive fluid in the direction of a positive, the potential function V, due to the exterior bodies, will here become V=-f.x-fcose.r=U.r. The finite integrals expressing the value of V reduce themselves therefore, in this case, to a single term, in which i = 1, and the corresponding value of D being We therefore see that the effect produced by the intervening shell, is to reduce the directive force which would act on a very small magnetic needle, In iron and other similar bodies, g is very nearly equal to 1, and therefore the directive force in the interior of a hollow spherical shell is greatly diminished, except when its thickness is very small compared with its radius, in which case, as is evident from the formula, it approaches towards the original value ƒ, and becomes equal to it when this thickness is infinitely small. To give an example of the use of the second formula, let it be proposed to determine the total action upon a point p, situate on one side of an infinitely extended plate of uniform thickness, when another point P, containing a unit of positive fluid, is placed on the other side of the same plate considering it as a perfect conductor of magnetism. For this, let fall the perpendicular PQ upon the side of the plate next P, on PQ prolonged, demit the perpendicular p, and make PQ=b, Pq=u, pq=v, and t = the thickness of the plate; then, since its action is evidently equal to that of an infinite sphere of the same thickness, whose centre is upon the line QP at an infinite distance from P, we shall have the required value of the total potential function at p by supposing a,a+t, a infinite, and the line PQ prolonged to be the axis from which the angle Now in the present case is measured and the value of the potential function, as before determined, is From the first expression we see that the general term U is a quantity of the order (a - b). Moreover, by substituting for r its value in u, (a — bij' ~+2 = (a − b)* (a − b + u)+2 = 1 e ̃ ̈" ; a a neglecting such quantities as are of the order compared with those retained. The general term U, and consequently U", ought therefore to be considered as functions of 7. a In the finite integrals just given, the increment of i is 1, and the corresponding increment of is 1 dy (because y dy (because a is in α = finite), the finite integrals thus change themselves into ordinary In fact (Méc. Cél. Liv. 3), U always integrals or fluents. satisfies the equation and as is infinitely small whenever V has a sensible value, we may eliminate it from the above by means of the equation. a = v, and we obtain by neglecting infinitesimals of higher orders than those retained, since 7, i a seeing that the remaining part of the general integral becomes infinite when v vanishes, and ought therefore to be rejected. It now only remains to determine the value of the arbitrary constant A. Making, for this purpose, = 0, i. e. v=0, we have By substituting for A and r their values, there results value ay, and neglecting infinitesimal quantities, we have dy. Writing now in the place of i its = where the integral relative to y is taken from y=0 to y∞, to correspond with the limits 0 and ∞o of i, seeing that i = ay. The preceding solution is immediately applicable to the imaginary case only, in which the inducing bodies reduce themselves to a single point P, but by the following simple artifice we may give it a much greater degree of generality: Conceive another point P', on the line PQ, at an arbitrary distance c from P, and suppose the unit of positive fluid concentrated in P' instead of P; then if we make r' = Pp, and Ø' =pPQ, we shall have u=r'cos e', v=r' sin e', and the value of the potential function arising from P' will be Moreover, the value of the total potential function at p due to this, arising from P and the plate itself, will evidently be obtained by changing u into u-c in that before given, and is therefore Expanding this function in an ascending series of the powers of which, as c is perfectly arbitrary, must be the part due to the term Q in the potential function arising from the inducing where the successive powers co, c2, c2, &c. of c are replaced by the arbitrary constant quantities k,, k,, k,, &c., the corresponding value of the total potential function will be given by making a like change in that due to P'. Hence if, for abridgement, we make the value of this function at the point p will be Now, if the original one due to the point P be called F, it is clear the expression just given may be written where the symbols of operation are separated from those of quantity, according to AKBOGAST's method; thus all the difficulty is reduced to the determination of F. Resuming therefore the original supposition of the plate's magnetic state being induced by a particle of positive fluid concentrated in P, the value of the total potential function at p will be |