Concave plate compared with flat one 93 



was spread uniformly in a ratio approaching much nearer to that Of 8 to one 

 than to that of equality. 



For if any particle of matter is removed from that part of AB which is 

 near C to that point which is at a distance from it, and an equal alteration is 

 made in the plate DF, the sum of the repulsions of these particles will be much 

 less after their removal than before. 



158] LEMMA XVII. Fig. 7. Let ACB be a thin plate, not flat but concave 

 on one side, let the radius of curvature of its surface be nowhere less than CV, 

 and let M V be perpendicular to its surface at C; let MC be very small in respect 

 of CV, and let Tt be a plane perpendicular to MC: the difference of the repulsion 



of any particle of matter as R in the plate ACB on the point M in the direction 

 CM, and of its repulsion on the point C in the same direction, is very nearly 

 the same as if the particle was transferred to T (CT being equal to the right 

 line CR), provided CR is small in respect of CV. 



Draw RN perpendicular to MC, the difference of the repulsions of R on the 



. _ MN CN MC CN CN ,._ 



points M and C = 3 - = 3 + - - 3 , and the difference of the 



MC 



but 



+ 



repulsions of the same particle placed at T on the same points = 

 MR* = (MC + CAT) 2 + RN 2 



= MC 2 + CR 2 + 2MC x CN 

 = MT 2 + 2MC x CN, 



CR 2 

 and CN is not greater than ~~ , and therefore zMC x CN is not greater than 



MC x CR 2 

 -^= -- , and therefore is very small in respect of CR 2 or MT 2 . 



O Y 



Therefore MR 2 differs very little from MT 2 , and TT from 

 This being premised there are two cases to be considered. 



