ELECTRICITY. 



535 



Theoretical 

 Electricity. 



AC 



X 



Fig. 10. 



sed by 3 n X AC X AC 3 -" AC X ~Q X 3 ' 



PLATE / 1 1 \ , 



DA*-"=3 nxACx AC X (AC^ ~ DA^lJ there - 



fore if a quantity of matter, which is to the whole quan- 

 tity in the plate as 2 AC to AC, be collected in the cir- 

 cumference, its repulsion on the column DC, will be to 



that of the whole plate as 3 X AC X AC x ( ~ r 



\ AC"- 1 



_ : , I to DC 3 ~ " + AC ! " DA 3 "; and con se- 



i.emmm 1 1 



quently the repulsion of the plate, when all the matter 

 is collected in its circumference, is to its repulsion when 



the matter is spread uniformly, as 



3 nx AC* 



X 



to 



Carol. I . If the length of the column is so great, that 

 AC" 1 is very small in respect to DC"', the repulsion 

 of the plate, when all the matter is collected in the cir- 

 cumference, is to its repulsion when the matter is spread 



uniformly, very nearly as -- - - to AC 3 ~ ", or 



as 3 n to 2. 



Co>ol. 2. If EC"- 1 is very small in respect of AC r >, 

 the repulsion of the plate on the short column EC, when 

 all the matter in the plate is collected in its circumfe- 

 rence, is to its repulsion when the matter is spread uni- 



formly, very nearly as 3 ~ "X" 1*EC' to EC 3 , 



" 



or as 3 X" 1 X EC" > to 4AC" ', and is there- 

 fore very small in comparison of what it is when the 

 matter is spread uniformly. 



For, by the same kind of process as was used in lem- 

 ma 8, it appears, that if EC* is very small in respect of 



AC 1 , then AC' X ( T7^_T TT^=T ) diflrers ver >' 



tie from 



-1 X EC' 



n lx EC' 



;andifEC"-i 



2AC" 



', then EC* is a fortiori 



.>* i 



is very small in respect of AC' 

 very small in respect of AC'. 



C',rol. 3. Suppose now that the matter of the plate is 

 denser near the circumference than near the middle, 

 and tliat the density at and near the middle is to the 

 mean density, or the density which it would every 

 where be of if the matter was spread uniformly, as J to 

 1, then the repulsion of the plate on EC will be le.-s than 

 if the matter was spread uniformly, in a ratio approach- 

 ing much nearer to that of 3 to 1, than to that of equa- 

 lity. 



Carol. 4. Let every thing be as in the last corollary, 

 and let * be taken to 1, as the force with which the 

 plate actually repels the column DC ( DC" 1 being very 

 great in respect of AC" ') is to the force with which it 

 would repel it, if the matter was spread uniformly ; the 

 repulsion of the plate on EC will be to its repulsion on 

 DC, in a ratio between that of EC'-" x 3 to AC S ~" x *, 

 and that of EC*-" to AC 3 "" x ,but will approach much 

 nearer to the former ratio than to the latter. 



Lemma 1 1. In the line DC produced, take CP equal 

 to CA : if all the matter of the plate AB is collected in 

 the circumference, its repulsion on the column CD, in- 



finitely continued, is equal to the repulsion of the same Theoretical 



quantity of matter collected in the point F, on the same Electricity^ 



column. For the repulsion of the plate on the column '"""Y"""' 



in the direction CD, is the same whether the matter of CCLI*" 



it be collected in the whole circumference, or in the fig, ig. 



point A. Suppose it therefore to be collected in A, 



and let an equal quantity of matter be collected in F ; 



take FG constantly equal to AD, and let AD and FG 



flow ; the fluxion of CD is to the fluxion of FG, as AD 



to CD ; and the repulsion of A on the point D, in the 



direction CD, is to the repulsion of F on G, as CD to 



AD ; therefore the fluxion of the repulsion of A on the 



column CD, in the direction CD, is equal to the fluxion 



of the repulsion of F on CG ; and when AD equals 



AC, the repulsion of both A and F on their respective 



columns vanishes ; and therefore the repulsion of A on 



the whole column CD, equals that of F on CG ; and 



when CD and CG are both infinitely extended, they 



may be considered as the same column. 



Prop. XV. Let two similar bodies, of different sizes, PROP. 13. 

 and consisting of different sorts of matter, be both over- 

 charged, or both undercharged, but in different de- 

 grees ; and let the redundance or deficience of fluid in 

 each be very small, in respect of the whole quantity of 

 fluid in them : it is impossible for the fluid to be dis- 

 posed accurately in a similar manner in both of them ;* 

 as it has been shown that there will be a space, close to 

 the surface, which will either be as full of fluid as it 

 can hold, or will be entirely deprived of fluid ; but it- 

 will be disposed as nearly in a similar manner in both 

 as is possible. To explain this, let DDE and bde, Fig. 

 12, be the two similar bodies; and let the space com- Fig. IS. 

 prehcnded between the surfaces BDE and FGH (or the 

 space BF as it may be called for shortness) be that part 

 of BDE, which is either as full of fluid as it can hold, 

 or entirely deprived of it : draw the surface ftfh, such 

 that the space If. shall be to the space BF, as the 

 quantity of redundant or deficient fluid in l/dc, to that 

 in BDE, and that the thickness of the space i/'shall 

 every where bear the same proportion to the' corre- 

 sponding thickness of BF : then will the space A/" be 

 either as full of fluid as it cmi hold, or entirely depri- 

 ved of it ; and the fluid within the space fgh will bs 

 disposed very nearly similarly to that in the space 

 FGH. 



For it is plain, that if the fluid could be disposed ac- 

 curately in a similar manner in botli bodies, the fluid 

 would be in equilibrio in one body, if it was in the 

 other ; therefore draw the surface 3; such that the 

 thickness of the space ftj, shall be every where to the 

 corresponding thickness of BF, as the diameter of Ids- 

 to the diameter of BDE ; and let the redundant fluid 

 or matter in /ifbe spread uniformly over the space 0f; 

 then if the fluid in the space fg/i is disposed exactly si- 

 milarly to that in FGII, it will be in equilibrio; as the. 

 fluid will then be disposed exactly similarly in tlie 

 spaces /SJi and BDE : but as, by the supposition, the 

 thickness of the space /3/"is very small in respect of the 

 diameter of bde, the fluid or matter in the space ///"will 

 exert very nearly the same force on the rest of the fluid, 

 whether it is spread over the space pj] or whether it is 

 collected in bj. 



Prop. XVI. Let two bodies, B and b, be connected PRO* 16. 

 to each other by a canal of any kind, and be either 

 over or under charged : it is plain that the quantity of 



Bpr the fluid being disposed in a umilai manner in both bodies, it meant tht the quantity of redundant or deficient fluid, in any small 

 part of one body, is to tl.at in the corresponding imall part of ihf other, as the whole quantity of redundant or deficient fluid in one body, 

 to that in the other, by the quantity of deficient fluid in a body, is meant the quantity of fluid wanting to saturate it. Thit xprcs. 

 sun it used, in order to avoid circumlocution. 

 3 



