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57]

CIRCULAR PLATE.

23

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.

A

E C

B

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

ᎠᏟ 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 CA :: CA: DA; the variable part of the fluent of

which is

- DC

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

the particle D is proportional to

ᎠᏟ

DO

(n − 1) DC"−1 — (n − 1) DA”-1 ›

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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 L11, h3” – L3¬" will be very small in respect of l3TM.

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which is very small in respect of is; as l1 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+ DC'3¬" — DA3¬" differs very little in this case from AC-"; and if DC is infinite, it is exactly equal to it.

62]

CIRCULAR RING.

25

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 EC to AC3

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

-n

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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 AC; the quantity of matter in the plate will be increased by a quantity, which is to the whole, as 2AĊ 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 2AC 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

when the matter is spread uniformly, very nearly as

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3-nx AC3-n

2

63] COR. II. If EC is very small in respect of AC", 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-n x n 1 × EC11 to 4AC-1; 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 EC is very small in respect of AC,

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differs very little from

n-1x EC2

› 2EA-1

or from

n-1x EC2
24C-1

; and if EC-1 is very small in respect of AC1, 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.

π

65] COR. IV. Let everything be as in the last corollary, and let 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 AC3-" × π, and that of EC to ACT, but will approach much nearer to the former ratio than to the latter.

67]

SIMILAR BODIES.

27

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.]

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