because =y and !=dy. Writing now in the place of i'its

1 a

value ay, and neglecting infinitesimal quantities, we have

where the integral relative to y is taken from y=0 to y = ∞, to correspond with the limits 0 and ∞ 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 O'pPQ, we shall have ur′ cos 0′, 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 c, the term multiplied by c' is

powers of

which, as c is perfectly arbitrary, must be the part due to the C1 in the potential function arising from the inducing

term Q+1

bodies. If then this function had been

Que le + qui le + Qm k + Qu k + &c.;

where the successive powers co, c', c', &c. of c are replaced by the arbitrary constant quantities ko, 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 ARBOGAST'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

8

dye-
— 9g3e ̃*

dB

F = 2 (1 − 9) (1 + 29) [ ̧ (2 +9) — 956*), √(1 – 5′′)

π

as was before shown.

Let

0

cos (Byv),

Writing now e3-1 in the place of cos (Byv), we obtain

provided we reject the imaginary quantities which may arise. In order to transform this double integral let

3g

the integral relative to z being taken from 0 to %=

2+9

The value of 1-g, for iron and other similar bodies, is very small; neglecting therefore quantities which are of the order (1-9) compared with those retained, there results

where u and v may have any values whatever provided they

F becomes by changing u into u+2t, we have

which, by effecting the integrations and rejecting the imaginary quantities, becomes

Suppose now po is a perpendicular falling from the point p upon the surface of the plate, and on this line, indefinitely extended in the direction Op, take the points P1, P, P,, &c., at the distances 2t, 4t, 6t, &c. from p; then F, F, F, &c. being the values of F, calculated for the points P1, P2 P3, &c. by the formula (a) of this article, and r', r',, r', &c. the corresponding values of r', we shall equally have

1 1

3r's

F= (1 − 9) (~ + + + &c. in infinitum);

seeing that F= 0.

From this value of F, it is evident the total action exerted upon the point p, in any given direction pn, is equal to the sum of the actions which would be exerted without the interposition of the plate, on each of the points p, P1, P., &c. in infinitum, in the directions pn‚ Ã ̧n, p‚”, &c. multiplied by the constant factor(1-g): the lines pn, p, p,n,, &c. being all parallel.

4

3

Moreover, as this is the case wherever the inducing point P

may be situate, the same will hold good when, instead of P, we substitute a body of any figure whatever magnetized at will. The only condition to be observed, is, that the distance between p and every part of the inducing body be not a very great quantity of the order

t

On the contrary, when the distance between p and the inducing body is great enough to render (1-9)r' a very con

t

siderable quantity, it will be easy to show, by expanding Fin a descending series of the powers of r', that the actions exerted upon p are very nearly the same as if no plate were interposed.

We have before remarked (art. 15), that when the dimensions of a body are all quantities of the same order, the results of the true theory differ little from those, which would be obtained by supposing the magnetic like the electric fluid, at liberty to move from one part of a conducting body to another; but when, as in the present example, one of the dimensions is very small compared with the others, the case is widely different; for if we make g rigorously equal to 1 in the preceding formulæ, they will belong to the latter supposition (art. 15), and as F will then vanish, the interposing plate will exactly neutralize the action of any magnetic bodies however they may be situate, provided they are on the side opposite the attracted point. This differs completely from what has been deduced above by employing the correct theory. A like difference between the results of the two suppositions takes place, when we consider the action exerted by the earth on a magnetic particle, placed in the interior of a hollow spherical shell, provided its thickness is very small compared with its radius, as will be evident by making g = 1 in the formulæ belonging to this case, which are given in a preceding part of the present article.

17. Since COULOMB's experiments on cylindric wires magnetized to saturation are numerous and very accurate, it was thought this little work could not be better terminated, than by directly deducing from theory such consequences as would admit of an immediate comparison with them, and in order to effect this, we