bodies the canal is inserted, or in whatever manner the two bodies are situated in respect of each other; provided that their distance is infinite, or so great that the repulsion of each body on the fluid in the canal shall not be sensibly less than if it was infinite.

Let the parallelograms AB and DF (Fig. 17) represent the two plates, and H and L the bodies communicating with them:

let now H be removed to h; and let it communicate with AB by the bent canal ge; the quantity of fluid in the plates and bodies remaining the same as before; and let us, for the sake of ease in the demonstration, suppose the canal gc to be everywhere of the same thickness as the canal GC; though the proposition will evidently hold good equally, whether it is or not the fluid will still be in equilibrio. For let us first suppose the canal gc to be continued through the substance of the plate AB, to C, along the line crC; the part crC being of the same thickness as the rest of the canal, and the fluid in it of the same density: by the preceding proposition, the repulsion or attraction of each particle of fluid or matter in the plates AB and DF, on the fluid in the whole canal Creg, in the direction of that canal, is equal to its repulsion or attraction on the fluid in the canal CG, in the

88]

BENT CANAL.

39

direction CG; and therefore the whole repulsion or attraction of the two plates on the canal Crcg, is equal to their repulsion or attraction on CG: but as the fluid in the plate AB is in equilibrio, each particle of fluid in the part Crc of the canal is impelled by the plates with as much force in one direction as the other; and consequently the plates impel the fluid in the canal cg with as much force as they do that in the whole canal Crcg, that is, with the same force that they impel the fluid in CG. In like manner the body h impels the fluid in cg with the same force that H does the fluid in CG; and consequently h impels the fluid in cg one way in the direction of the canal, with the same force that the two plates impel it the contrary way; and therefore the fluid in cg has no tendency to flow from one body to the other.

87] COR. By the same method of reasoning, with the help of the corollary to the 23rd proposition, it appears, that if AB and H each communicate with a third body by canals of incompressible fluid, and a communication is made between AB and H by another canal of incompressible fluid, the fluid will have no tendency to flow from one to the other through this canal; supposing that the fluid was in equilibrio before this communication was made. In like manner if AB and H communicate with each other, or each communicate with a third body, by canals of real fluid, instead of the imaginary canals of incompressible fluid used in these propositions, and a communication is also made between them by a canal of incompressible fluid, the fluid can have no tendency to flow from one to the other. The truth of the latter part of this corollary will appear by supposing an imaginary canal of incompressible fluid to be continued through the whole length of the real one.

88] PROP. XXV. Let now a communication be made between the two plates AB and DF, by the canal NRS of incompressible fluid, of any length; and let the body H and the plate AB be overcharged. It is plain that the fluid will flow through that canal from AB to DF. Now the whole force with which the fluid in the canal is impelled along it by the joint action of the two plates is the same with which the whole quantity of fluid in the canal CG or cg is impelled by them; supposing the canal NRS to be everywhere of the same breadth and thickness. as CG or cg.

For suppose that the canal NRS, instead of communicating with the plate DF, is bent back just before it touches it, and continued infinitely along the line Ss; the force with which the two plates impel the fluid in Ss, is the same with which they impel that in EL, supposing Ss to be of the same breadth and thickness as EL; and is therefore nothing; therefore the force with which they impel the fluid in NRS, is the same with which they impel that in NRSS; which is the same with which they impel that in CG.

89] PROP. XXVI. Let now xyz [Fig. 17] be a body of an infinite size, containing just fluid enough to saturate it; and let a communication be made between h and xyz, by the canal hy of incompressible fluid, of the same breadth and thickness as gc or GC; the fluid will flow through it from h to xyz; and the force with which the fluid in that canal is impelled along it, is equal to that with which the fluid in NRS is impelled by the two plates.

If the canal hy is of so great a length, that the repulsion of h thereon is the same as if it was continued infinitely, then the thing is evident: but if it is not, let the canal hy, instead of communicating with xyz, so that the fluid can flow out of the canal into xyz, be continued infinitely through its substance, along the line yu now it must be observed that a small part of the body xyz, namely, that which is turned towards h, will by the action of h upon it, be rendered undercharged; but all the rest of the body will be saturated; for the fluid driven out of the undercharged part will not make the remainder, which is supposed to be of an infinite size, sensibly overcharged: now the force with which the fluid in the infinite canal hyv is impelled by the body h and the undercharged part of xyz, is the same with which the fluid in gc is impelled by them; but as the fluid in all parts of xyz is in equilibrio, a particle in any part of yv cannot be impelled in any direction; and therefore the fluid in hy is impelled with as much force as that in hyv; and therefore the fluid in hy is impelled with as much force as that in gc; and is therefore impelled with as much force as the fluid in NRS is impelled by the two plates.

90] It perhaps may be asked, whether this method of demonstration would not equally tend to prove that the fluid in hy was impelled with the same force as that in NRS, though xyz did not

93]

CANAL OF REAL FLUID.

41

contain just fluid enough to saturate it. I answer not; for this demonstration depends on the canal yv being continued, within the body xyz, to an infinite distance beyond any over or undercharged part; which could not be if xyz contained either more or less fluid than that*.

91] PROP. XXVII. Let two bodies B and b (Fig. 13) be joined by a cylindric or prismatic canal Aa, filled with real fluid; and not by any imaginary canal of incompressible fluid as in the 20th proposition; and let the fluid therein be in equilibrio: the force with which the whole or any given part of the fluid in the canal is impelled in the direction of its axis by the united repulsions and attractions of the redundant fluid or matter in the two bodies and the canal, must be nothing; or the force with which it is impelled one way in the direction of the axis of the canal, must be equal to that with which it is impelled the other way.

For as the canal is supposed cylindric or prismatic, no particle of fluid therein can be prevented from moving in the direction of the axis of it, by the sides of the canal; and therefore the force with which each particle is impelled either way in the direction of the axis, by the united attractions and repulsions of the two bodies and the canal, must be nothing, otherwise it could not be at rest; and therefore the force with which the whole, or any given part of the fluid in the canal, is impelled in the direction of the axis, must be nothing.

92] COR. I. If the fluid in the canal is disposed in such manner, that the repulsion or attraction of the redundant fluid or matter in it, on the whole or any given part of the fluid in the canal, has no tendency to impel it either way in the direction of the axis; then the force with which that whole or given part is impelled by the two bodies must be nothing; or the force with which it is impelled one way in the direction of the axis, by the body B, must be equal to that with which it is impelled in the contrary direction by the other body; but not if the fluid in the canal is disposed in a different manner.

93] COR. II. If the bodies, and consequently the canal, is overcharged; then, in whatever manner the fluid in the canal is disposed, the force with which the whole quantity of redundant fluid in the canal is repelled by the body B in the direction Aɑ,

[* Note 5.]

must be equal to that with which it is repelled by b in the contrary direction. For the force with which the redundant fluid is impelled in the direction Aa by its own repulsion, is nothing; for the repulsions of the particles of any body on each other have no tendency to make the whole body move in any direction.

94] REMARKS. When I first thought of the 20th and 22nd propositions, I imagined that when two bodies were connected by a cylindric canal of real fluid, the repulsion of one body on the whole quantity of fluid in the canal, in one direction, would be equal to that of the other body on it in the contrary direction, in whatever manner the fluid was disposed in the canal; and that therefore those propositions would have held good very nearly, though the bodies were joined by cylindric canals of real fluid; provided the bodies were so little over or undercharged, that the quantity of redundant or deficient fluid in the canal should be very small in respect of the quantity required to saturate it; and consequently that the fluid therein should be very nearly of the same density in all parts. But from the foregoing proposition it appears that was mistaken, and that the repulsion of one body on the fluid in the canal is not equal to that of the other body on it, unless the fluid in the canal is disposed in a particular manner : besides that, when two bodies are both joined by a real canal, the attraction or repulsion of the redundant matter or fluid in the canal has some tendency to alter the disposition of the fluid in the two bodies; and in the 22nd proposition, the canal CG exerts also some attraction or repulsion on the canal EM: on all which accounts the demonstration of those propositions is defective, when the bodies are joined by real canals. I have good reason however to think, that those propositions actually hold good very nearly when the bodies are joined by real canals; and that, whether the canals are straight or crooked, or in whatever direction the bodies are situated in respect of each other: though I am by no means able to prove that they do: I therefore chose still to retain those propositions, but to demonstrate them on this ideal supposition, in which they are certainly true, in hopes that some more skilful mathematician may be able to shew whether they really hold good or not. [See Note 3.]

95] What principally makes me think that this is the case, is that as far as I can judge from some experiments I have made*, [* Exp. III., Art. 265.]