Equidistant concave plates 



89 



153] PROP. XXXIII. If two bodies B and b are successively connected by 

 canals of incompressible fluid to a third body C placed at an infinite distance 

 from them, and are overcharged, that is, if one of them, as B, is first con- 

 nected to G and afterwards B is removed and b put in its room, the quantity 

 of redundant fluid in C being the same in both cases, it is plain that the quantity 

 of redundant fluid in B will bear the same proportion to that in b that it would 

 if B and b were placed at an infinite distance from each other, and connected 

 by canals of incompressible fluid. 



154] LEMMA XV. Fig. 5. Let AB be a thin flat plate of any shape what- 

 soever, of uniform thickness and composed of uniform matter. Let CG be an 

 infinitely slender cylindric column of uniform matter perpendicular to the plane 

 of AB and meeting it in C and extended infinitely beyond G. Let ab be a thin 

 circular plate perpendicular to cG whose center is c. Let the area of ab be equal 

 to that of AB, and let the quantity of matter in it be the same, and let it be 

 disposed uniformly. 



Fig. 5- 



Let B be that point of the circumference of AB which is nearest to C. If 

 EC is small in respect of CB, the repulsion of the plate AB on the short column 

 EC is to the repulsion of ab on the infinite column cG nearly as EC to cb. 



For let BD be a circle drawn through B with center C, as EC is very small 

 in respect of CB, the repulsion of the circle BD on EC is to its repulsion on CG 

 very nearly as EC to CB, and therefore is to the repulsion of ab on cG very 

 nearly as EC to cb. But the repulsion of AB on EC is very little greater than 

 that of DB, for the repulsion of DB is very near as great as it would be if its 

 size was infinite. 



155] LEMMA XVI. {Fig. 6.} Let ACB and DEF be two thin plates, not flat 

 but concave on one side, let their distance be everywhere the same, and let it be 

 very small in respect of the radius of curvature of all parts of their surface. 

 Let C be any point of the surface of AB, and let CE be perpendicular to the 

 surface in that point. Let Tt be a flat plate perpendicular to CE. 



Let R be any point in AB and S the corresponding point in DF, and let T 



