Comparison of similar bodies 55 



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 pris- 

 matic canal A a, filled with incompres- 

 sible fluid, in the same manner as '' l ' " p 



described in the preceding proposition : 

 let the bodies be overcharged; 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 A a; 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 i, and 

 less than 3, the quantity of redundant fluid in the two bodies will be to 

 each other as the n i 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 a/ 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 -j-~ to > ; for, as the fluid is disposed similarly in both bodies, 

 A r aj 



the quantity of fluid in any small part of B, is to the quantity in the 

 corresponding part of b, as AF"- 1 to a/"" 1 ; and consequently the repulsion 

 of that small part of B, on D, is to the repulsion of the corresponding 



1 i i 



part of b, on d, as . F , or --}-=. , to -, . But the quantity of fluid in the 

 AF" AF af 



small part DE of the 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 A a; for, by 

 the supposition, the repulsion of B on A a 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. 



