66] LEMMA XI. In the line DC produced, take CF equal to CA: if all the matter of the plate AB is collected in the circumference, its repulsion on the column CD, infinitely continued, is equal to the repulsion of the same quantity of matter collected in the point F, on the same column.

For the repulsion of the plate on the column in the direction CD, is the same, whether the matter of it be collected in the whole circumference, or in the point A. Suppose it therefore to be collected in A; and let an equal quantity of matter be collected in F; take FG constantly equal to AD; and let AD and FG flow: the fluxion of CD is to the fluxion of FG, as AD to CD; and the repulsion of A on the point D, in the direction CD, is to the repulsion of F on G, as CD to AD; and therefore the fluxion of the repulsion of A on the column CD, in the direction CD, is equal to the fluxion of the repulsion of F on CG; and when AD equals AC, the repulsion of both A and F on their respective columns vanishes; and therefore the repulsion of A on the whole column CD equals that of F on CG; and when CD and CG are both infinitely extended, they may be looked upon as the same column.

67] PROP. XVII. Let two similar bodies, of different sizes, and consisting of different sorts of matter, be both overcharged, or both undercharged, but in different degrees; and let the redundance or deficience of fluid in each be very small in respect of the whole quantity of fluid in them: it is impossible for the fluid to be disposed accurately in a similar manner in both of them*; as it has been shewn that there will be a space, close to the surface, which will either be as full of fluid as it can hold, or will be entirely deprived of fluid; but it will be disposed as nearly in a similar manner in both, as is possible. To explain this, let BDE and bde (Fig. 11) be the two similar bodies; and

By the fluid being disposed in a similar manner in both bodies, I mean that the quantity of redundant or deficient fluid in any small part of one body, is to that in the corresponding small part of the other, as the whole quantity of redundant or deficient fluid in one body, to that in the other. By the quantity of deficient fluid in a body, I mean the quantity of fluid wanting to saturate it. Notwithstanding the impropriety of this expression, I must beg leave to make use of it, as it will frequently save a great deal of circumlocution. [See Note 1.]

let the space comprehended between the surfaces BDE and FGH (or the space BF as I shall call it for short

ness) be that part of BDE, which is either as full of fluid as it can hold, or entirely deprived of it: draw the surface fgh, such that the space bf shall be to the space BF, as the quantity of redundant or deficient fluid in

bde, to that in BDE, and that the thickness of the space bf shall everywhere bear the same proportion to the corresponding thickness of BF: then will the space bf be either as full of fluid as it can hold, or entirely deprived of it; and the fluid within the space fgh will be disposed very nearly similarly to that in the space FGH

For it is plain, that if the fluid could be disposed accurately in a similar manner in both bodies, the fluid would be in equilibrio in one body, if it was in the other: therefore draw the surface Bde, such that the thickness of the space Bf shall be everywhere to the corresponding thickness of BF, as the diameter of bde to the diameter of BDE; and let the redundant fluid or matter in bf be spread uniformly over the space f; then if the fluid in the space fgh is disposed exactly similarly to that in FGH, it will be in equilibrio; as the fluid will then be disposed exactly similarly in the spaces ẞde and BDE: but as by the supposition, the thickness of the space Bf is very small in respect of the diameter of bde, the fluid or matter in the space bf will exert very nearly the same force on the rest of the fluid, whether it is spread over the space Bf, or whether it is collected in bf.

68] PROP. XVIII. Let two bodies, B and b, be connected to each other by a canal of any kind, and be either over or undercharged: it is plain that the quantity of redundant or deficient fluid in B, would bear exactly the same proportion to that in b, whatever sort of matter B consisted of, if it was possible for the redundant or deficient fluid in any body to be disposed accurately in the same manner, whatever sort of matter it consisted of. For suppose B to consist of any sort of matter; and let the fluid in the canal and two bodies be in equilibrio: let now B be made to consist of some other sort of matter, which requires a different

quantity of fluid to saturate it; but let the quantity and disposition of the redundant or deficient fluid in it remain the same as before: it is plain that the fluid will still be in equilibrio; as the attraction or repulsion of any body depends only on the quantity and disposition of the redundant and deficient fluid in it. Therefore, by the preceding proposition, the quantity of redundant or deficient fluid in B, will actually bear very nearly the same proportion to that in b, whatever sort of matter B consists of; provided the quantity of redundant or deficient fluid in it is very small in respect of the whole. [See Exp. IV., Art. 269.]

69] PROP. XIX. Let two bodies B and b (Fig. 12) be con

b

Fig. 12.

B

nected together by a very slender canal ADda, either straight or crooked: let the canal be everywhere of the same breadth and thickness; so that all sections of this canal made by planes perpendicular to the direction of the canal in that part, shall be equal and similar: let the canal be composed of uniform matter; and let the electric fluid therein be supposed incompressible, and of such density as exactly to saturate the matter therein; and let it, nevertheless, be able to move readily along the canal; and let each particle of fluid in the canal be attracted and repelled by the matter and fluid in the canal and in the bodies B and b, just in the same manner that it would be if it was not incompressible*; and let the bodies B and b be either over or undercharged. I say that the force with which the whole quantity of fluid in the canal is impelled from A towards D, in the direction of the axis of the canal, by the united attractions and repulsions of the two bodies, must be nothing; as otherwise the fluid in the canal could not be at rest: observing that by the force with which the whole quantity of fluid is impelled in the direction of the axis of the canal, I mean the sum of the forces, with which the fluid in each part of the canal is impelled in the direction of the axis of the canal in that place,

*This supposition of the fluid in the canal being incompressible, is not mentioned as a thing which can ever take place in nature, but is merely imaginary; the reason for making of which will be given hereafter.

from A towards D; and observing also, that an impulse in the contrary direction from D towards A must be looked upon as negative.

For as the canal is exactly saturated with fluid, the fluid therein is attracted or repelled only by the redundant matter or fluid in the two bodies. Suppose now that the fluid in any section of the canal, as Ee, is impelled with any given force in the direction of the canal at that place, the section Dd would, in consequence thereof, be impelled with exactly the same force in the direction of the canal at D, if the fluid between Ee and Dd was not at all attracted or repelled by the two bodies; and, consequently, the section Dd is impelled in the direction of the canal, with the sum of the forces, with which the fluid in each part of the canal is impelled by the attraction or repulsion of the two bodies in the direction of the axis in that part; and consequently, unless this sum was nothing, the fluid in Dd could not be at rest.

70] COR. Therefore, the force with which the fluid in the canal is impelled one way in the direction of the axis, by the body B, must be equal to that with which it is impelled by b in the contrary direction.

71] PROP. XX. Let two similar bodies B and b (Fig. 13) be connected by the very slender cylindric or prismatic canal Aa,

filled with incompressible fluid,

in the same manner as described in the preceding proposition: let the bodies be overcharged;

Fig. 13.

but let the quantity of redundant fluid in each bear so small a proportion to the whole, that the fluid may be considered as disposed in a similar manner in both; let the bodies also be similarly situated in respect of the canal Aa; and let them be placed at an infinite distance from each other, or at so great an one, that the repulsion of either body on the fluid in the canal shall not be sensibly less than if they were at an infinite distance: then, if the electric attraction and repulsion is inversely as the n power of the distance, n being greater than 1, and less than 3, the quantity of redundant fluid in the two bodies will be to each other as the n-1 power of their corresponding diameters AF and af.

For if the quantity of redundant fluid in the two bodies is in this proportion, the repulsion of one body on the fluid in the canal will be equal to that of the other body on it in the contrary direction; and, consequently, the fluid will have no tendency to flow from one body to the other, as may thus be proved. Take the points D and E very near to each other; and take da to DA, and ea to EA, as af to AF; the repulsion of the body B on a particle at D, will be to the repulsion of b on a particle at d,

as

1

1

to ; for, as the fluid is disposed similarly in both bodies, AF af'

the quantity of fluid in any small part of B, is to the quantity in the corresponding part of b, as AF" to af-1; and consequently the repulsion of that small part of B, on D, is to the AF-1 repulsion of the corresponding part of b, on d, as

1

1

or AF

AF

to But the quantity of fluid in the small part DE of the af

canal, is to that in de, as DE to de, or as AF to af; therefore the repulsion of B on the fluid in DE, is equal to that of b on the fluid in de therefore, taking ag to Aa, as af to AF, the repulsion of b on the fluid in ag, is equal to that of B on the fluid in Aa; but the repulsion of bon ag may be considered as the same as its repulsion on Aa; for, by the supposition, the repulsion of B on Aa may be considered as the same as if it was continued infinitely; and therefore, the repulsion of b on ag may be considered as the same as if it was continued infinitely.

N.B. If n was not greater than 1, it would be impossible for the length of Aa to be so great, that the repulsion of B on it might be considered as the same as if it was continued infinitely; which was my reason for requiring n to be greater than 1.

72] COR. By just the same method of reasoning it appears, that if the bodies are undercharged, the quantity of deficient fluid in b will be to that in B, as af" to AF-1.

73] PROP. XXI. Let a thin flat plate be connected to any other body, as in the preceding proposition, by a canal of incompressible fluid, perpendicular to the plane of the plate; and let that body be overcharged, the quantity of redundant fluid in the plate will bear very nearly the same proportion to that in the