248 PHILOSOPHICAL TRANSACTIONS. [aNNO 1771 



cutting AE in b and (3, and af in d and S; and draw the radii rb and pd. As 

 PB = PD, the force with which p repels a particle at b, in the direction bj3, is to 

 that with which it repels an equal particle at d, in the direction ni, as 



— to — i, or as —- to -5,; and therefore the force with which it repels the whole 

 fluid in Bj3, in the direction Bi3, is the same with which it repels the whole fluid 

 in dS, in the direction dJ', that is in the direction of the canal; and therefore the 

 force with which it repels the whole fluid in ae, in the direction ae, is the same 

 with which it repels the whole fluid in af, in the direction of the canal. 



Corol. If the bent canal adf, instead of being infinitely continued, meets 

 the cylindric canal in E, as in fig. 16, the repulsion of p on the fluid in the 

 bent canal ade, in the direction of the canal, will still be equal to its repulsion 

 en that in the cylindric canal ae, in the direction ae. 



Prop. 24. If two bodies, for instance the plate ab, and the body h, of prop. 

 22, communicate with each other, by a canal filled with incompressible fluid, 

 and are either over or undercharged; the quantity of redundant fluid in them 

 will bear the same proportion to each other, whether the canal by which they 

 communicate is straight or crooked, or into whatever part of the 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 dp, 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 gc; 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 every where of the same thickness as 

 the canal gc; though the proposition will evidently hold equally good, whether 

 it is or not: then 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 arc ; 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 crag, in the direction of that canal, is equal to its repul- 

 sion or attraction on the fluid in the canal cg, in the direction cg; and there- 

 fore the whole repulsion or attraction of the two plates on the canal crag, 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 ia 



