Theory of a coated plate 95 



Let these coatings be of any shape whatsoever, and let their edges correspond 

 as in Lemma XVI, Cor. I. 



Let AB communicate with the body H, and DF with the body L, by the 

 straight canals CG and EM of incompressible fluid. 



Let the points C and E be so placed that the two canals shall form one right 

 line perpendicular to A B at the point C, and let the lengths of these canals be so 



w 



71 



Fig. 8. 



great that the repulsion of the coatings on the fluid in them shall be not sensibly 

 less than if they were infinite, and let H be overcharged and let L be saturated. 



It is plain from Prop. XII that DF will be undercharged, and that AB will 

 be more overcharged than it would otherwise be. 



Let Ww be a thin flat circular plate whose center is C, perpendicular to CE, 

 and whose area is equal to that of AB, let the force with which the redundant 

 fluid in AB would repel the short column CE (if ME was continued to C) be 

 called m, and let the force with which it would repel CM, or with which it repels 

 CG (for they are both alike), be called M. Let the force with which the same 

 quantity of redundant fluid disposed in DF, in the same manner in which the 



deficient fluid therein is actually disposed, would repel \p r \ be called * , let 



the force with which the same quantity of redundant fluid uniformly disposed 

 on Ww would repel CG be called W, and let the force with which H repels CG 

 be the same with which a quantity of fluid, which we will call B, uniformly 

 distributed on Ww would repel it in the contrary direction: then will the 



GW 

 quantity of redundant fluid in AB be B x ,-j ~ , which, if M and G 



Mg + Gm mg 



are very nearly alike, and m and g are very small in respect of G, differs very 



BW 



little from - , and the deficient fluid in DF will be to the redundant fluid 

 g+ m 



in AB as M m to G, and therefore on the same supposition will be very 

 nearly equal to it. 



For the force with which AB repels the fluid in EM must be equal to that 

 with which DF attracts it, for otherwise some fluid would run out of DF into L, 

 or out of L into DF. For the same reason the excess of the repulsion of AB 

 on CG above the attraction of DF thereon must be equal to the force with which 

 a quantity of redundant fluid equal to B spread uniformly on Ww would repel it. 



