VbL. LXI.] PHILOSOPHICAL TKANSACTIONS. 230 



of uniform matter ; and let the diameter of the column be so small, that the re- 

 pulsion of the plate ab on it shall not be sensibly different from what it would 

 be, if all the matter in it was collected in the axis : the force with which the 

 plate repels the column, is proportional to dc^'" + ac^'" — DA*""; supposing 

 the thickness of the plate and base of the eolumn to be given. !For, if dc is 

 supposed to flow, the corresponding fluxion of the repulsion is proportional to 



D'C DC X D'C DX D'A ^, /J . /• 1 • 1 AC'"" +DC'~" — DA*"" 



— T-^ T^- = — z—. — — T-,; the fluent of which, ,va- 



»c— * da" ' DC' ■ DA"-' ' 3 — n ' ,, 



nishes when do vanishes. 



Carol. 1. If the length of the column is so great, that AC"''-is very small in 

 respect of do""', the repulsion of the plate on it is very nearly the same as if the 

 column was infinitely continued. For, by lemma 8, ac^"" + dc'"" — - da'"" dif- 

 fers very little in this case from ac'""; and if dc is infinite, it is exactly equal 

 to it. ^ .' ." 



Corol. 1. If AC""' is very small in respect of dc""', and the point e be taken 

 in DC, such that ec""' shall be very small in respect of ac""', the repulsion of 

 the plate on the small part of the column ec, is to its repulsion on the whole 

 column DC, very nearly as ec'"" to ac'"". 



Lemma 10. If we now suppose all the matter of the plate to be collected in 

 the circumference of the circle, so as to form an infinitely slender uniform ring, 

 its repulsion on the column dc will be less than when the matter is spread uni- 

 formly all over the plate in the ratio of 2lf|jl£' x (7^ — ^^) to dc'"" + 



AC""" — da'"". n'>V'>'f;. . 



For it was before said, that if the matter of the plate be spread uniformly, its 

 repulsion on the column will be proportional to dc'"" -f- ac'"" — da'~", or 

 may be expressed by it; let now ac, the semidiameter of the plate, be in- 

 creased by the infinitely small quantity a'C ; the quantity of matter in the 

 plate will be.increased by a quantity, which is to the whole, as 2a-c to ac ; and 

 the repulsion of the plate on the column will be increased by 3 — n X A'C X 



ac'""— a-c X — X 3 — n X da^"" = 3 — ra X A'C X ac X (— — ; —): 



DA '■AC""' da" ••' 



therefore if a quantity of matter, which is to the whole quantity in the plate, 

 as 2a*c to AC, be collected in the circumference, its repulsion on the column dc, 



will be to that of the whole plate, as 3 — ra X a-c x ac X ( — ;;^, ;^,) to 



dc'"" -|- ac'~" — da'"" ; and consequently 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 ^ ~ " ^ ^""^ X {—— — :^^) to dc'"" -|- ac'"" — da'"". 

 Corol. 1. If the length of the column is so great, that ac" ~ ' is very small 



