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, Fig. 13. de ED В a 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 AF af to ; for, as the fluid is disposed similarly in both bodies, the quantity of fluid in any small part of B, is to the quantity in the corresponding part of b, as AF-1 to af"; and consequently the repulsion of that small part of B, on D, is to the repulsion of the corresponding part of b, on d, as 1 AF-1 1 AF' to 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 b on 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. But the quantity of fluid in the small part DE of the 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"1 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 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 and therefore, if P is very small, will be equal to B 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 with which a quantity equal to B π , spread in the manner in which 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 x 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 x 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 − A × (1 − P)2, or A × (2P - P2), spread over AB, in the manner in which the redundant fluid therein is actually spread, B would repel it therefore A× (2P - P2) must be equal to π B must be equal to 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-1 is very small in respect of AC-1, and that EC3-" is very small in respect of AC", the quantity of redundant В АС3 fluid in AB will be greater than X B AC 2 EC , and less than X , but will approach much nearer to the latter value 28 EC than the former. For, in this case, Pπ is, by Lemma X. Corol. IV., less than and greater than EC EC 3-n x 8, but approaches AC much nearer to the latter value than the former; and if EC13is 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 |