50 First published Paper on Electricity 



3-n / 72\3-n 



- 



For 



therefore 



/ 72\3 



/ j + 1. V 



' 



3-n_ L 3-n (3 - ") * 2 3~Xn-IX/ 



- 1 ~ ~~ ' 6 



-" x 3 x /"~ 1 / 3 ~" X3 x ix /"+ 1 

 ""- 1 1 ~ ' 



which is very small in respect of I 3 -"; as I"- 1 is by the supposition very 

 small in respect of L"- 1 . 



58] LEMMA IX. Let DG now represent the axis of a cylindric or pris- 

 matic column of uniform matter; and let the diameter of the column be 

 so small, that the repulsion 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 3 ~" + AC 3 ~" DA 3 ~"', 

 supposing the thickness of the plate and base of the column to be given. 



For, if DC is supposed to flow, the corresponding fluxion of the re- 

 pulsion is proportional to 



DC DC x DC DC DA 

 DC"~ 2 "DA"- 1 ~~ DC"-* ~ DA"~ Z ' 



AC 3 ~" + DC 3 ~ n DA 3 ~" 

 the fluent of which, - - , vanishes when DC vanishes. 



59] COR. I. If the length of the column is so great that AC"- 1 is very 

 small in respect of DC"- 1 , the repulsion of the plate on it is very nearly 

 the same as if the column was infinitely continued. 



For by Lemma VIII AC 3 ~" + DC 3 -" - DA 3 ~" differs very little in 

 this case from AC 3 ~"; and if DC is infinite, it is exactly equal to it. 



60] COR. II. If AC"- 1 is very small in respect of DC"- 1 , and the 

 point E be taken in DC such that EC"- 1 shall be very small in respect 

 of AC"- 1 , 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 3 ~" to AC 3 -". 



61] LEMMA X. 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 uniformly all over the plate, in the ratio of 



(3-n)AC\ 



2 



