CHAP. I.] SUPPLEMENT TO CHAP. XIV. 279 



If the piece had been originally straight, dA<j> would be equal to d $, and 



-5 ^ = , , . = -; hence from (13) we have M = . 



a s r d (p r f 



Prom the calculus we have the radius of curvature 

 r~ fd y\ ~i 



L 1 + feOJ *<* 



r = \ or approximately, r = -55-; 



<Ta? 



<Z a v 

 hence M = E I ji (13 &) 



This is the equation assumed in the Supplement to Chap. XUL, Art. 2. 



2. Displacement of any point. We indicate the horizontal dis- 

 placement of any point of the axis along the axis of x by A x, along y by 

 A y. The corresponding changes of d x, d y, and d s are A d x, Ady, Ads. 

 The total horizontal displacement is then dx + Adx = (da + Ads) co9 

 (0 + A 0) . The total vertical displacement [sdy+Ady = (d8 + Ads)B\n 

 (9 + A 0). Hence, since A d x = d A x, 



cos ($ + A 0) d x, 

 d A y = (d s+ A ds) sin (< + A 0) dy. 



By Trigonometry, cos ($ -f A 0) = cos <f> cos A < sin $ sin A <p, 

 sin (<^ + A <) = sin cos A < + cos sin A </>, or if cos A $ = 1, 



d! <Z y 



sin A < = A 9, cos 9 = -p , sin = -= . 



tt S G, S 



d x dill 



J7 -A0^, 



dy 

 gf 



Substituting these in the equations above, 



/ Ad\ 

 = (d!a! A<j>dy) ( 1 +~j 1 ^ 



(Ads\ 

 1 + -J-) ^y, 



or, removing the parentheses, and neglecting quantities of the second order 



with respect to A and 5 , 

 d s 



Ads , 

 -j dx 



Ads , 

 j ay 



