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over there will be a certain space, as AabB, adjoining to the plane AB, in which the fluid will be pressed close together; and the fluid in that space will press against the plane AB, and will endeavour to escape from it; and by Prop. II. the two bodies will attract each other: now I say that the force with which the fluid presses against the plane AB, is very nearly the same with which the two bodies attract each other in the direction EA; provided that no part of AEFB is undercharged.

Suppose so much of the fluid in each part of the cylinder as is sufficient to saturate the matter in that part, to become solid; the remainder, or the redundant fluid remaining fluid as before. In this case the pressure against the plane AB must be exactly equal to that with which the two bodies attract each other in the direction EA: for the force with which D attracts that part of the fluid which we supposed to become solid, is exactly equal to that with which it repels the matter in the cylinder; and the redundant fluid in EabF is at liberty to move, if it had any tendency to do so, without moving the cylinder; so that the only thing which has any tendency to impel the cylinder in the direction EA is the pressure of the redundant fluid in AabB against AB; and as the part near EF is saturated, there is no redundant fluid to press against the plane EF, and thereby to counteract the pressure against AB. Suppose now all the electric fluid in the cylinder to become fluid; the force with which the two bodies attract each other will remain exactly the same; and the only alteration in the pressure against AB, will be, that that part of the fluid in AabB, which we at first supposed solid and unable to press against the plane, will now be at liberty to press against it; but as the density of the fluid when its particles are pressed close together may be supposed many times greater than when it is no denser than sufficient to saturate the matter in the cylinder, and consequently the quantity of redundant fluid in Aab B many times greater than that which is required to saturate the matter therein, it follows that the pressure against AB will be very little more than on the first supposition.

N.B. If any part of the cylinder is undercharged, the pressure against AB is greater than the force with which the bodies attract. If the electric repulsion is inversely as the square or some higher power of the distance, it seems very unlikely that any part of the

cylinder should be undercharged; but if the repulsion is inversely as some lower power than the square, it is not improbable but some part of the cylinder may be undercharged.

55] LEMMA VII. Let AB (Fig. 10) represent an infinitely thin flat circular plate, seen edgeways,

so as to appear to the eye as a straight line; let C be the center of the circle; and let DC passing through C, be perpendicular to the plane of the plate; and let the plate be of uniform thickness, and consist of uniform matter, whose particles repel with a force inversely as the n power of the distance; n being greater than one, and

Fig. 10.

E C

F

less than three: the repulsion of the plate on a particle at D is pro

DC

DC

portional to DC-DA-1; provided the thickness of the plate and size of the particle D is given.

For if CA is supposed to flow, the corresponding fluxion of the quantity of matter in the plate is proportional to CA × CÀ; and the corresponding fluxion of the repulsion of the plate on the particle D, in the direction DC, is proportional to

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for DÅ : CẢ :: CA: DA; the variable part of the fluent of

which is

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(n-1) DA-1 whence the repulsion of the plate on

the particle D is proportional to

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(n − 1) DC”−1 − (n − 1) DA"−1 ›

56] COR. If DC-1 is very small in respect of CA", the particle D is repelled with very nearly the same force as if the diameter of the plate was infinite.

57] LEMMA VIII. Let L and represent the two legs of a right-angled triangle, and h the hypothenuse; if the shorter leg is so much less than the other, that 7-1 is very small in respect of L"-1, h3-" — L3-" will be very small in respect of l3-".

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which is very small in respect of "; as 1 is by the supposition very small in respect of L

58] LEMMA IX. Let DG now represent the axis of a cylindric or prismatic column of uniform matter; and let the diameter of the column be so small, that the repulsion of the plate AB on it shall not be sensibly different from what it would be, if all the matter in it was collected in the axis: the force with which the plate repels the column is proportional to

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supposing the thickness of the plate and base of the column to be given.

For, if DC is supposed to flow, the corresponding fluxion of the repulsion is proportional to

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59] COR. I. If the length of the column is so great that AC is very small in respect of DC, the repulsion of the plate on it is very nearly the same as if the column was infinitely continued.

For by Lemma VIII. AC+ DC3-" - DA3-" differs very little in this case from AC"; and if DC is infinite, it is exactly equal to it.

60] COR. II. If AC is very small in respect of DC-1, and the point E be taken in DC such that EC shall be very small in respect of AC-1, the repulsion of the plate on the small part of the column EC, is to its repulsion on the whole column DC, very nearly as EC3" to AC-n

61] LEMMA X. If we now suppose all the matter of the plate to be collected in the circumference of the circle, so as to form an infinitely slender uniform ring, its repulsion on the column DC will be less than when the matter is spread uniformly all over the plate, in the ratio of

(3-n) AC2 2

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For it was before said, that if the matter of the plate be spread uniformly, its repulsion on the column will be proportional to DC3¬" + AC3¬” — DA3-", or may be expressed thereby; let now AC, the semidiameter of the plate, be increased by the infinitely small quantity AĊ; the quantity of matter in the plate will be increased by a quantity, which is to the whole, as 2AČ to AC; and the repulsion of the plate on the column will be increased by

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therefore if a quantity of matter, which is to the whole quantity in the plate as 240 to AC be collected in the circumference, its repulsion on the column DC will be to that of the whole plate

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and consequently the repulsion of the plate when all the matter is collected in its circumference, is to its repulsion when the matter is spread uniformly, as

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62] COR. I. If the length of the column is so great, that A C"-1 is very small in respect of DC-1, the repulsion of the plate, when

all the matter is collected in the circumference, is to its repulsion 3-nx AC3-n

when the matter is spread uniformly, very nearly as 2

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63] COR. II. If EC-1 is very small in respect of AC"-1, the repulsion of the plate on the short column EC, when all the matter in the plate is collected in its circumference, is to its repulsion when the matter is spread uniformly, very nearly as

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or as 3-nxn-1 x EC-1 to 4AC"; and is therefore very small in comparison of what it is when the matter is spread uniformly.

For by the same kind of process as was used in Lemma VIII., it appears, that if EC2 is very small in respect of AC3,

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differs very little from

n-1x EC2
or from
>
2EA"-1

n-1× EC2
2ACn-1

; and if EC-1 is very small in respect of AC-1, EC is à fortiori very small in respect of AC.

64] COR. III. Suppose now that the matter of the plate is denser near the circumference than near the middle, and that the density at and near the middle is to the mean density, or the density which it would everywhere be of if the matter was spread uniformly, as 8 to 1; the repulsion of the plate on EC will be less than if the matter was spread uniformly, in a ratio approaching much nearer to that of 8 to 1, than to that of equality.

65] COR. IV. Let everything be as in the last corollary, and let π be taken to one, as the force with which the plate actually repels the column DC, (DC-1 being very great in respect of AC-1), is to the force with which it would repel it, if the matter was spread uniformly; the repulsion of the plate on EC will be to its repulsion on DC, in a ratio between that of EC-" × 8 to AC13-" × π, and that of EC-" to AC3-x, but will approach much nearer to the former ratio than to the latter.

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