ELECTRICITY. 



537 



Pi ITL 

 ' 



ii.-l plates; let the thickness of the plates be very small in 

 tltctrkiry. re spect of their distance CE ; let the plate AB com- 

 s ~~ ' municate with the body H, and the plate DF with the 

 body L, by the canals CG and EM of incompressible 

 fluid, such as are described in Prop. 17 ; let these ca- 

 nals meet their respective plates in their centres C and 

 E, and be perpendicular to the plane of the plates ; 

 and let their length be so great, that the repulsion of 

 the plates on the fluid in them may be considered as 

 the same as if they -were cbntinued infinitely ; let the 

 body H be overcharged, and let L be saturated. It is 

 plain, from Prop. 10, that DF will be undercharged, and 

 AB will be more overcharged than it would otherwise 

 be. Suppose now, that the redundant fluid in AB is 

 disposed m the same manner as the deficient fluid is in 

 DF ; let P be to 1, as the force with which the plate 

 AB would repel the fluid in CE, if the canal ME was 

 continued to C, is to the force with which it would 

 repel the fluid in CM ; and let the force with which 

 AB repels the fluid in CG, be to the force with which 

 it would repel it, if the redundant fluid in it was 

 spread uniformly, as it to 1 ; and let the force with 

 which the body H repels the fluid in CG, be the same 

 with which a quantity of redundant fluid, which we 

 will call B, spread unifonnly over AB, would repel it 

 in the contrary direction. Then will the redundant 



fluid in AB be equal to np^pT- > aml therefore, if 



T> 



P is very small, will be very nearly equal to g-p^-; 



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 rea- 

 son, the excess of the repulsion of AB on the fluid in 

 CG, 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 



1> 

 to , spread in the manner in which the redundant fluid 



is actually spread in AB, would repel it. By the gup- 

 position, the force with which AB repels the fluid in 

 I.M. 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 flukl in CM, is the same with 

 which an equal quantity similarly disposed in DF, 

 would repel the fluid in EM ; therefore the force with 

 which die redundant fluid in AB repels the fluid in 

 I.M, is to that witli which an equal quantity similarly 

 disposed in DF, would repel it, as 1 P to 1 : there- 

 lore, 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 A x ( I P ) ', to A ; therefore, the excess of tlie 

 force with which AB repels CG, above that with 

 which DF attracts it, is equal to that with which a 

 quantity of redundant fluid equal to A A x ( 1 P)', 

 orAx(2P I"), spread over A B, in the manner in 

 which the redundant fluid in it is actually spread. 

 would repel it: therefore, Ax('^P P') must be equal 



B B 



to , or A must be equal to ^ - JJT-. 



v 6i T*r ?r 



VOL. Till. PART II. 



Carol. 1. If the density of the redundant fluid near TheoretfraS 

 the middle of the plate AB is less than the mean den- 

 sity, or the density which it would every where be 

 of, if it was spread uniformly, in the ratio of 3 to 1 ; 

 and if the distance of the two plates is so small, that 

 EC" 1 is very small in respect of AC"', and that ' 

 EC 3 " is very small in respect of AC 3 ", the quanti- 

 ty of redundant fluid in AB will be greater than 



o V 1** C* I * 



approach much nearer to the latter value than the former.. 

 For, in this case, PIT is, by lemma 10, corol. 4, less 



(Ec\ 3 * /EC\ S " 



- 1 and greater than ( v^ ) X but ap* 

 AC/ \AW 



proaches much nearer to the latter value than the for- 

 mer; and if EC 3 ~" is very small in respect of AC 3 -", 

 P is very small. 



Remarks. If DF was not undercharged, it is cer- 

 tain that AB would be considerably more overcharged 

 near the circumference of the circle than near the 

 centre ; for if the fluid was spread uniformly, a parti- 

 cle placed any where at a distance from the centre, a* 

 at N, would be repelled with considerably more force 

 towards the circumference, than it would towards the 

 centre. If the plates are very near together, and con- 

 sequently DF nearly as much undercharged as AB is 

 overcharged, AB will still be more overcharged near 

 the circumference than near the centre, but the dif- 

 ference will not be near so great as in the former case : 

 for, letNRbe many times greater than CE, and NS 

 less than CE ; and take Er and Es equal to CR and 

 (S; there can be no doubt, he thinks, but that the 

 deficient fluid in DF will be lodged nearly in the same 

 manner as the redundant fluid in AB"; und therefore 

 the repulsion of the redundant fluid at H, on a particle 

 at N, will be rery nearly balanced by the attraction of 

 the redundant matter at r, for R is not much nearer to 

 N than r is ; but the repulsion of S will not be near 

 balanced by that of*; for the distance of S from N is 

 much less than that of *. Let now a small circle, 

 whose diameter is ST, be drawn round the centre N, 

 on the plane of the plate ; as the density of the fluid is 

 greater at T than at S, the repulsion of the redundant 

 fluid within the small circle tends to impel the point 

 N towards I ; but as there is a much greater quantity 

 of fluid between N and B, fhan between N and A, the 

 repulsion of the fluid without the small circle tends to 

 balance that; but the effect of the fluid within the 

 small circle is not much less than it would be, if DF 

 was not undercharged ; whereas much the greater part 

 of the effect of that part of the plate on the outside of 

 the circle, is taken off by the effect of the correspond- 

 ing part of DF: consequently the difference of density 

 between T and S will not be near so great, as if DF 

 was not undercharged. Hence it is probable, that if 

 the two plates are very nearly together, the density o 

 the redundant fluid near the centre will not be much 

 less than the mean density, or 3 will not be much less 

 than 1 ; moreover, the less the distance of the plates, 

 the nearer will 3 approach to I . 



Cvrol. 2. Let now the body H consist of a circular 

 plate, of the same size as AB, placed so that the canal 

 CG shall pass through its centre, and be perpendicular 

 to its plane ; by the supposition the force with which 

 H repels the fluid in the canal CG, is the same with, 

 which a quantity of fluid, equal to B, spread unifonnly 

 over AB, would repel it in a contrary direction : there- 

 fore, if the fluid in the plate H was spread uniformly, 

 3 Y 



