Ratios of capacities 83 



and fD (cf being taken equal to cC), supposing that the redundant fluid in all 

 three plates is disposed in the same manner. 



For first, as the plates AB and ab are at an infinite distance from any other 

 over or undercharged body, the repulsion of A B on the canal Cc in one direction 

 must be equal to that of ab on it in the contrary, and therefore the redundant 

 fluid in AB must be equal to that in ab. 



Secondly, the sum of the repulsions of AB and ab on the canal cD must be 

 equal to that of EF on it in the contrary direction, as otherwise some fluid 

 must flow from ab to EF or from EF to ab. But as all three plates are of the 

 same size, and the fluid in them is disposed in the same manner, the repulsions 

 of EF and ab on cD will be to each other as the quantity of redundant fluid in 

 them, and therefore the quantity of redundant fluid in ab will be to that in EF 

 as the repulsion of ab on CD to the sum of the repulsions of AB and ab on it, 

 that is, as the repulsion of ab on cD to the sum of its repulsions on fD and cD, 

 for the repulsion of AB on cD is equal to the repulsion of ab on/D*. 



142] COR. I. If the fluid in these plates is disposed in the same manner as 

 in Prop. XXIX the quantity of redundant fluid in each of the plates AB and ab 

 will be to that in EF as 



AC(p+\) to ACQ + V+p (Ac - Cc) + 



For by Lemma X the repulsion of a given quantity of fluid spread uniformly 

 over ab on the column cD; the repulsion of the same fluid on cf; the repulsion 

 of the same quantity of fluid collected in the circumference of the plate ab on 

 the column cD ; and the repulsion of the same fluid on cf are to each other as 



, ac , ac ac 2 

 ac; ac + cf af; and >, 



2 2 2af 



and therefore the whole repulsion of the plate ab on cD is to its repulsion on cf as 



ac ... ac ac 2 



pxac+ 2 : p x (ac + cf- /) + -_ ^ 



and therefore the repulsion of ab on cD is to the sum of its repulsions on cD 

 and/Z) as 



acx(p + l):ac(2p+i)-p (ac + cf- af) - + ?L, 



ac z 

 or as ac(p + $:ac(p + \}+p(af-cf) + j. 



143] COR. II. Therefore if all the redundant fluid in the plates is spread 

 uniformly, the redundant fluid in each of the plates AB and ab will be to that 

 in EF as AC : AC + Ac Cc, and if it is all collected in the circumference, as 



AC* 



AC :AC + A ~~. 

 Ac 



144] COR. III. By Prop. XXIV it appears that the redundant fluid in the 

 plate AB or ab will bear the same proportion to that in EF though they com- 



[* Note ii, p. 377.] 



62 



