240 PHILOSOPHICAL TRANSACTIONS. [aNNO 1771. 



in respect of DC*- ', the repulsion of the plate when all the matter is 

 collected in the circumference, is to its repulsion when the matter is spread 



uniformly, very nearly as ^~"^ -^^ to ac^ - «, or as 3 — m to 2. 



Corol. 1. If EC" - ■ is very small in respect of AC ~ ', the repulsion of the 



plate on the short column ec, when all the matter in the plate is collected in its 



circumference, is to its repulsion when the matter is spread uniformly, very nearly as 

 3— n X n — 1 X Ec' „ , . 

 T^rr to Ec3 - ", or as 3 — K X n — 1 X eC - ' to 4ac' - ' ; and is 



therefore very small in comparison of what it is when the matter is spread 



uniformly, w :;iiij (h,. 



- For, by the isame kind of process as was used in lemma 8, it appears, that if 



EC" is very small in respect of ac^ then ac' X (- „_ , — — 7tri)difFersverylittle 



from^-^^^ — ■""> or from V^rr"^ ^^^ '^ ec' ~ ■ is very small in respect of 



AC"^,- ', then EC^ is a fortiori very small in respect of ac". 



Carol. 3. Suppose now that the matter of the plate is denser near the circum- 

 ference than near the middle, and that 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 less 

 than if the matter was spread uniformly, in a ratio approaching much nearer to 

 that of ^ to 1, than to that of equality. 



Corol. 4. Let every thing be as in the last corollary, and let tt be taken to ] , 

 as the force with which the plate actually repels the column dc (dc' ~ ' 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 do, in a ratio between that of ecJ - " x S to ac^ - » x t, and 

 that of ECJ - " to AcJ - " X T, but will approach much nearer to the former 

 ratio than to the latter. 



Lemma 1 1. In the line dc produced, take cf equal to ca: if all the matter of 

 the plate ab is collected in the circumference, its repulsion on the column cd, 

 infinitely continued, is equal to the repulsion of the same quantity of matter col- 

 lected in the point f, on the same column. For the repulsion of the plate on the 

 column in the direction CD, is the same whether the matter of it be collected in 

 the whole circumference, or in the 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 



