238 PHILOSOPHICAL TRANSACTIONS. [aNNO 1771. 



the matter in the cylinder, and consequently the quantity of redundant fluid in 

 Aa/)B many times greater than that which is required to saturate the matter in 

 it, it follows that the pressure against ab will be very little more than on the first 

 supposition. 



n. b. If any part of the cylinder is undercharged, the pressure against ab is 

 greater than the force with which the bodies attract. If the electric repulsion is 

 inversely as the square or some higher power of the distance, it seems very un- 

 likely that any part of the cylinder should be undercharged; but if the repulsion 

 is inversely as some lower power than the square, it is not improbable but some 

 part of the cylinder may be undercharged. 



Lemma 7- LetAB, fig. 10, represent an infinitely thin flat circular plate, seen 

 edgeways, so as to appear to the eye as a straight line; let c be the centre of the 

 circle; and let DC, passing through c, be perpendicular to the plane of the plate; 

 and let the plate be of uniform thickness, and consist of uniform matter, whose 

 particles repel with a force inversely as the n power of the distance ; n being 

 greater than 1 , and less than 3 : the repulsion of the plate on a particle at d, is 



DC DC 



proportional to — -.-in — — ;rri ; provided the thickness of the plate and size of 

 the particle d is given. 



For if CA is supposed to flow, the corresponding fluxion of the quantity of 

 matter in the plate, is proportional to ca X c-a; and the corresponding fluxion 

 of the repulsion of the plate on the particle d, in the direction dc, is propor- 



, . CA X C-A .^ DC DA X DC ^ . ^ ..ill 



tional to ;; — X — , = z — ; for d'a is to ca :: CA : da; *he variable 



DA" da' da" ' 



part of the fluent of which is _ j:;:;^ : whence the repulsion ot the plate on 



the particle d, is proporti6nal to 



DC DC .DC DC 



-, or to 



n — 1 X DC"""' »— 1 X DA"~' DC""' da"~' 



Carol. If DC"-' is very small in respect of ca"-', the particle d is repelled 

 with very nearly the same force as if the diameter of the plate was infinite. 



Lemma. Let l and / represent the two legs of a right angled triangle, and h 

 the hypothenuse; if the shorter leg / is so much less than the other, that /""' is 

 very small in respect of l"-', then A^-" — \^- " will be very small in respect of 



li-". For A'-" = (l= + P)^ = L'-" X (1 + -y~ - l'-" X 1 + ^-^—^ 

 _ 3-„ X n- I X ^_^ &c. therefore A- ^'-" - 3-» x/'^3-„ x « - . x /' 



8l* ' 2l"-' 8l" + ' 



- /'-"x3-nx^' />-" X 3-no< «- 1 X /" + ■ a u- u • n 



Kc. = —— ; &c. which IS very small 



21."-' 8i,«+' •' 



in respect of /*""; as /"-' is by the supposition very small in respect of l""'. 

 Lemma Q. — Let do now represent the axis of a cylindric or prismatic column 



