ρ

and being the densities of the electric fluid at the surfaces A and B respectively. Permitting ourselves, in what follows, to neglect quantities of the order compared with those retained, it is clear that we may write for 0, and hence by substitution

B' B

where V and p are quantities of the order; and ẞ being

for any point on the sur

the order or unity. The only thing which now remains to be determined, is the value of face A.

d2 V
dw2

Throughout the substance of the glass, the potential function V will satisfy the equation 0=8V, and therefore at a point on the surface of A, where of necessity w, w' and w" are each equal to zero, we have

the horizontal mark over w, w' and w" being, for simplicity, omitted. Then since w' = 0,

and as is constant and equal to at the surface A, there hence arises

R being the radius of curvature at the surface A, in the plane (w, w). Substituting these values in the expression immediately preceding, we get

In precisely the same way we obtain, by writing R for the radius of curvature in the plane (w, w0′′),

both rays being accounted positive on the side where w, i. e. w, is negative. These values substituted in 0=8V, there results

1

+
R R

for the required value of and thus the sum of the two

=

therefore the required values of the densities p and p are

which values are correct to quantities of the order op, or, which is the same thing, to quantities of the order ; these having been neglected in the latter part of the preceding analysis, as unworthy of notice.

Suppose do is an element of the surface A, the corresponding element of B, cut off by normals to A, will be

do

{1+0+), and therefore the quantity of fluid on this

R R

1

last element will be pdo {1 ++); substituting for ō its

R'

value before found, P=-P{1-0+), and neglecting

op, we obtain

-pdo,

(

the same quantity as on the element do of the first surface. If, therefore, we conceive any portion of the surface A, bounded by a closed curve, and a corresponding portion of the surface B, which would be cut off by a normal to A, passing completely round this curve; the sum of the two quantities of electric fluid, on these corresponding portions, will be equal to zero; and consequently, in an electrical jar any how charged, the total quantity of electricity in the jar may be found, by calculating the quantity, on the two exterior surfaces of the metallic coatings farthest from the glass, as the portions of electricity, on the two surfaces adjacent to the glass, exactly neutralise each other. This result will appear singular, when we consider the immense quantity of fluid collected on these last surfaces, and moreover, it would not be difficult to verify it by experiment.

As a particular example of the use of this general theory: suppose a spherical conductor whose radius a, to communicate with the inside of an electrical jar, by means of a long slender wire, the outside being in communication with the common reservoir; and let the whole be charged: then P representing the density of the electricity on the surface of the conductor, which will be very nearly constant, the value of the potential function within the sphere, and, in consequence of the communication established, at the inner coating A also, will be 4πaP very nearly, since we may, without sensible error, neglect the action of the wire and jar itself in calculating it. Hence

B=4πаP and B' = 0,

and the equations (8), by neglecting quantities of the order 0, give

We thus obtain, by the most simple calculation, the values of

the densities, at any point on either of the surfaces A and B, next the glass, when that on the spherical conductor is known.

The theory of the condenser, electrophorous, &c. depends upon what has been proved in this article; but these are details into which the limits of this Essay will not permit me to enter; there is, however, one result, relative to charging a number of jars by cascade, that appears worthy of notice, and which flows so readily from the equations (8), that I cannot refrain from introducing it here.

Conceive any number of equal and similar insulated Leyden phials, of uniform thickness, so disposed, that the exterior coating of the first may communicate with the interior one of the second; the exterior one of the second, with the interior one of the third; and so on throughout the whole series, to the exterior surface of the last, which we will suppose in communication with the earth. Then, if the interior of the first phial be made to communicate with the prime conductor of an electrical machine, in a state of action, all the phials will receive a certain charge, and this mode of operating is called charging by cascade. Permitting ourselves to neglect the small quantities of free fluid on the exterior surfaces of the metallic coatings, and other quantities of the same order, we may readily determine the electrical state of each phial in the series: for thus, the equations (8) become

Designating now, by an index at the foot of any letter, the number of the phial to which it belongs, so that, p, may belong to the first, P, to the second phial, and so on; we shall have, by supposing their whole number to be n, since is the same for every one,

Now B represents the value of the total potential function, within the prime conductor and interior coating of the first phial, and in consequence of the communications established in this system, we have in regular succession, beginning with the prime conductor, and ending with the exterior surface of the last phial, which communicates with the earth,

But the first system of equations gives 0=p,+ Par P+ P., whatever whole numbers may be, and the second line of that just exhibited is expressed by 0=P11+ P.; hence by comparing these two last equations,

=

which shows that every phial of the system is equally charged. Moreover, if we sum up vertically, each of the columns of the first system, there will arise in virtue of the second

We therefore see, that the total charge of all the phials is precisely the same, as that which one only would receive, if placed in communication with the same conductor, provided its exterior coating were connected with the earth. Hence this mode of charging, although it may save time, will never produce a greater accumulation of fluid than would take place if one phial only were employed.

(9.) Conceive now a hollow shell of perfectly conducting matter, of any form and thickness whatever, to be acted upon by any electrified bodies, situate without it; and suppose them to