Charge on circular ring 5 1 



For it was before said, that if the matter of the plate be spread uni- 

 formly, its repulsion on the column will be proportional to 



DC 3 -" + AC s ~ n - DA 3 - n , 



or may be expressed thereby ; let now A C, the semidiameter of the plate, 

 be increased by the infinitely small quantity AC; 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 



AC 

 (3 - n) AC x AC 2 "' - AC x x (3 - n) x DA 2 ~, 



= (3 - n) x A C x AC x 



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

 plate as 2 A C to A C, be collected in the circumference, its repulsion on the 

 column DC will be to that of the whole plate as 



3 - n x AC x AC x ^ - , , to DC*-" + AC*~" - DA 3 ~"; 







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 uni- 

 formly, as 



3 ~ n x . y _ L- -J: 

 2 



( _ L- -J: ^ to DC 3 ~" 4- AC 3 -" DA 3 ~ n 

 UO- 1 DA"- 1 ) ' 



62] 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, when all the matter 

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



o _ M 

 spread uniformly, very nearly as - - to AC 3 ~ n , or as 3 n to 2. 



63] COR. II. If EC n ~ l is very small in respect of AC"~ l , 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 ix EC 2 , 



. ,. to EC 3 " 1 , 



qAC"- 1 



oras3 n x n ix EC"- 1 to ^AC"- 1 ; and is therefore 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 Lemma VIII, it appears, 

 that if EC 2 is very small in respect of AC 2 , 



* (1C""" 1 ~ EA 



n i x EC 2 n i x EC 2 



differs very little from - . - , or from - -17* - ', and if EC"~ l 



2EA"' 1 2AC n ~ l 



is very small in respect of AC"- 1 , EC 2 is a fortiori very small in respect 

 of AC 2 . 



42 



