VOL. LXI.] PHILOSOPHICAL TRANSACTIONS. 245 



over AB, would repel it in the contrary' direction. Then will the redundant 

 fluid in AB be equal to - — ~— J ^^^ therefore, if p is very small, will be very 



nearly equal to — ; 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 repul- 

 sion of AB on the fluid in co, above the attraction of fd on it, 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 -, 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 quan- 

 tity 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 l : 

 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 X 1 — p, or ax 

 (l — >)*, 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 redun- 

 dant fluid equal to a — A X (1 — p)% or a X (2p — p^), spread over ab, in 

 the manner in which the redundant fluid in it is actually spread, would repel it : 

 therefore, a X (2p — p*) must be equal to -, or a must be equal to . 



Corol. 1. 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 every where be of, if it 

 was spread uniformly, in the ratio of J' to 1 ; and if the distance of the two plates 

 is so small, that ec""' is very small in respect of ac""', and that ec'~" is very 

 small in respect of ac'~", the quantity of redundant fluid in ab will be greater than 

 |- X (— )^"", and less than ~ X (^)^"''^ but will approach much nearer to the 

 latter value than the former. For, in this case, ptt is, by lemma 10, corol. 4, 

 less than (—)^~', and greater than ( — Y'" X <?, but approaches much nearer to 



AC AC 



the latter value than the former ; and if ec*~" is very small in respect of ac*"", p 

 is very sm;ill. 



