9 o 



Preliminary Propositions 



be the corresponding point in Tt*: the sum of the repulsions of R on the column 

 CE in the direction CE and of S on the same column in the opposite direction 

 EC is very nearly equal to the force with which they would repel the same 

 column in the direction CE if they were both transferred to T, provided CR* 



Fig. 6. 







is very small in respect of the square of the least radius of curvature of the 

 surface of AB. 



Let RS be continued till it meets CE continued in V, draw EM and SN 

 perpendicular to CR. 



Let CM = C,RE-RM = E, SC - NC = S, and SE - NM = D. 



As CE is very small in respect of the least radius of curvature of AB, and 

 CV is not less than the least radius of curvature, CM and NR are each very 

 small in respect of CR, and therefore CN, MR, and ES differ from CR in a 

 very small ratio. Moreover as CR 2 is very small in respect of CF 2 , CM 2 and 

 RN 2 are very small in respect of CE 2 , and therefore ME and NS differ in a very 



CE 2 

 small ratio from CE ; and, moreover, 2 x (TE TC) is greater than ==- . 



Now the repulsion of the point R on the column CE in the direction CE is 



_ A jffr f?r* 



r = ^7; , and the repulsion of the point 5 on the same column 



RC RE RC x RE 



SC SE 

 in the opposite direction is ^^r ~p-, and the sum of the repulsions of R 



oO X o jC- 



and S is 



RE-RC SC -SE E-C 



RC x RE 



SC x SE 

 E 



RC x RE 



S 



S + C-D 



SC x SE' 

 D 



C 



C 



~ RC x RE ^ SC x SE ^ SC x SE RC x RE ^ SC x SE' 



* If RS is drawn perpendicular to the surface of A B at the point R cutting DF 

 in S, I call S the corresponding point of the plate DF, and if CT is taken in the 

 intersection of the plane RCE with that of the plate Tt equal to the right line CR, 

 I call 2" the corresponding point of Tt. 



f Lemma XII [Art. 145]. 



