other body, whatever the thickness of the plate may be, provided its thickness is very small in proportion to its breadth, or smallest diameter.

For there can be no doubt, but what, under that restriction, the fluid will be disposed very nearly in the same manner in the plate, whatever its thickness may be; and therefore its repulsion on the fluid in the canal will be very nearly the same, whatever its thickness may be. [See Exp. IV., Art. 272.]

74] PROP. XXII. Let AB and DF (Fig 14) represent two equal and parallel circular plates, whose centres are C and E; let

the plates be placed so, that a right line joining their centers shall be perpendicular to the plates; let the thickness of the plates be very small in respect of their distance CE; let the plate AB communicate with the body H, and the plate DF with the body L, by the canals CG and EM of incompressible fluid, such as are described in Prop. XIX; let these canals meet their respective plates in their centers C and E, and be perpendicular to the plane of the plates; and let their length be so great, that the repulsion of the plates on the fluid in them may be considered as the same as if they were continued infinitely; let the body H be overcharged, and let L be saturated. It is plain, from Prop. XII., that DF will be undercharged, and AB will be more overcharged than it would otherwise be. Suppose, now, that the redundant fluid in AB is disposed in the same manner as the deficient fluid is in DF; let P be to one as the force with which the plate AB would repel the fluid in CE, if the canal ME was continued to C, is to the force with which it would repel the fluid in CM; and let the force with which AB repels the fluid in CG, be to the force with which it would

repel it, if the redundant fluid in it was spread uniformly, as π to 1; and let the force with which the body H repels the fluid in CG, be the same with which a quantity of redundant fluid, which we will call B, spread uniformly over AB, would repel it in the contrary direction. Then will the redundant fluid in AB B and therefore, if P is very small, will

be equal to

2Pπ – Р2Ã'

be very nearly equal to

B

2P and the deficient fluid in DF will

be to the redundant fluid in AB, as 1-P to 1, and therefore, if P is very small, will be very nearly equal to the redundant fluid in AB.

For it is plain, that the force with which AB repels the fluid in EM, must be equal to that with which DF attracts it; for otherwise, some fluid would run out of DF into L, or out of L into DF: for the same reason, the excess of the repulsion of AB on the fluid in CG, above the attraction of FD thereon, must be equal to the force with which a quantity of redundant fluid equal to B, spread uniformly over AB, would repel it, or it must be equal to that

the redundant fluid is actually spread in AB, would repel it. By the supposition, the force with which AB repels the fluid in EM, is to the force with which it would repel the fluid in CM, supposing EM to be continued to C, as 1-P to 1; but the force with which any quantity of fluid in AB would repel the fluid in CM, is the same with which an equal quantity similarly disposed in DF, would repel the fluid in EM; therefore the force with which the redundant fluid in AB repels the fluid in EM, is to that with which an equal quantity similarly disposed in DF, would repel it, as 1-P to 1: therefore, if the redundant fluid. in AB be called A, the deficient fluid in DF must be A×1-P: for the same reason, the force with which DF attracts the fluid in CG, is to that with which AB repels it, as A×1-P×1-P, or Ax (1-P), to A; therefore, the excess of the force with which AB repels CG above that with which DF attracts it, is equal to that with which a quantity of redundant fluid equal to A-Ax (1P), or Ax (2P - P'), spread over AB, in the

manner in which the redundant fluid therein is actually spread,

would repel it therefore A × (2P – P2) must be equal to

B

π

or A

75] COR. I. If the density of the redundant fluid near the middle of the plate AB, is less than the mean density, or the density which it would everywhere be of, if it was spread uniformly, in the ratio of 8 to 1; and if the distance of the two plates is so small, that EC is very small in respect of AC"1, and that EC is very small in respect of AC", the quantity of redundant B AC3-n

fluid in AB will be greater than X

B AC 28 X EC

3-n

,

2 EC

2

and less than

but will approach much nearer to the latter value

than the former. For, in this case, Pπ is, by Lemma X. Corol. IV.,

less than

EC3-n

AC

and greater than

EC
AC

3

x8, but approaches much nearer to the latter value than the former; and if EC3¬n is very small in respect of AC", P is very small.

76] REMARKS. If DF was not undercharged, it is certain that AB would be considerably more overcharged near the circumference of the circle than near the center; for if the fluid was spread uniformly, a particle placed anywhere at a distance from the center, as at N, would be repelled with considerably more force towards the circumference than it would towards the center. If the plates are very near together, and, consequently, DF nearly as much undercharged as AB is overcharged, AB will still be more overcharged near the circumference than near the center, but the difference will not be near so great as in the former case: for, let NR be many times greater than CE, and NS less than CE; and take Er and Es equal to CR and CS, there can be no doubt, I think, but that the deficient fluid in DF will be lodged nearly in the same manner as the redundant fluid in AB; and therefore, the repulsion of the redundant fluid at R, on a particle at N, will be very nearly balanced by the attraction of the redundant matter at r, for R is not much nearer to N than r is; but the repulsion

of S will not be near balanced by that of s; for the distance of S from N is much less than that of s. Let now a small circle, whose diameter is ST, be drawn round the center N, on the plane of the plate; as the density of the fluid is greater at T than at S, the repulsion of the redundant fluid within the small circle tends to impel the point N towards C; but as there is a much greater quantity of fluid between N and B, than between N and A, the repulsion of the fluid without the small circle tends to balance that; but the effect of the fluid within the small circle is not much less than it would be, if DF was not undercharged; whereas much the greater part of the effect of that part of the plate on the outside of the circle, is taken off by the effect of the corresponding part of DF: consequently, the difference of density between T and S will not be near so great as if DF was not undercharged. Hence I should imagine, that if the two plates are very near together, the density of the redundant fluid near the center will not be much less than the mean density, or 8 will not be much less than 1; moreover, the less the distance of the plates, the nearer will & approach to 1.

77] COR. II. Let now the body H consist of a circular plate, of the same size as AB, placed so, that the canal CG shall pass through its center, and be perpendicular to its plane; by the supposition, the force with which H repels the fluid in the canal CG, is the same with which a quantity of fluid, equal to B, spread uniformly over AB, would repel it in the contrary direction: therefore, if the fluid in the plate H was spread uniformly, the quantity of redundant fluid therein would be B, and if it was all collected 2B in the circumference, would be 3-ni

and therefore the real

quantity will be greater than B, and less than

2B

3-n'

78] COR. III. Therefore, if we suppose & to be equal to 1, the quantity of redundant fluid in AB will exceed that in the

it appears that the real quantity of redundant fluid in AB can hardly be much greater than it would if & was equal to 1.

79] COR. IV. Hence, if the electric attraction and repulsion is inversely as the square of the distance, the redundant fluid in AB, supposing & to be equal to 1, will exceed that in the plate H, in a greater ratio than that of AC to 4CE, and less than that of AC to 2CE.

80] COR. V. Let now the body H consist of a globe, whose diameter equals AB; the globe being situated in such a manner, that the canal CG, if continued, would pass through its center; and let the electric attraction and repulsion be inversely as the square of the distance, the quantity of redundant fluid in the globe will be 2B: for the fluid will be spread uniformly over the surface of the globe, and its repulsion on the canal will be the same as if it was all collected in the center of the sphere, and will therefore be the same with which an equal quantity, disposed in the circumference of AB, would repel it in the contrary direction, or with which half that quantity, or B, would repel it, if spread uniformly over the plate. [See Art. 140.]

81] COR. VI. Therefore, if 8 was equal to 1, the redundant. fluid in AB would exceed that in the globe, in the ratio of AC to 4CE; and therefore, it will in reality exceed that in the globe, in a rather greater ratio than that of AC to 4CE; but if the plates are very near together, it will approach very near thereto, and the nearer the plates are, the nearer it will approach thereto.

82] COR. VII. Whether the electric repulsion is inversely as the square of the distance or not, if the body H is as much undercharged, as it was before overcharged, AB will be as much undercharged as it was before overcharged, and DF as much overcharged as it was before undercharged.

83] COR. VIII. If the size and distance of the plates be altered, the quantity of redundant or deficient fluid in the body H remaining the same, it appears, by comparing this proposition with the 20th and 21st propositions, that the quantity of redundant and 3-n AC EC

deficient fluid in AB will be as AC-1 ×

posing the value of 8 to remain the same *.

or as >

AC2
?

ECз- Sup

,

[* Note 4.]