CHAP. H.] SUPPLEMENT TO CHAP. XTV. 285 



(5) Horizontal displacement. 



According to eq. (14), Art. 2, since a = rsin0, y = r(l cos0), 

 <Z a? = r cos <Z 0, d y = r sin $ d 0, we have 



ElAaj= r / [H r 9 (sin cos a) V r* (0 sin a + cos 0), 



K r* (H cos a + V sin a) + A \ sin d 0, 



* r 3 / (H cos a + V sin a) cos <j> d 0. 

 The integration gives us the three equations, 



E I A z = r* I H ( sin cos cos a sin + cos a cos 0) 



V (sin a sin sin a cos + | sin 9 0) I 



K r* (H cos a + V sin a) cos A r (1 cos 0) + B. 



E I A ' = r* I H (^ ^ sin cos cos a sin -f cos a cos 0) 



V (sin a sin sin a cos i sin" 0) I 



K r* (H oosa + V sin a) cos A' r (1 cos 0) + B'. 



E I A* = r 8 1 H ($ -J sin cos cos a sin + cos a cos 0) 



V (sin a sin sin a cos i sin* 0) I 



K r 3 (H cos a + V sin a) cos A' r (1 cos 0) + B'. 



For = /3, that is at the load, A x must equal A x'. 

 Hence, after reduction, 



B-B' = -iPr 1 (2 + sin^-2cosj3-2j3sin/3) + /cPr 8 ^sin^ . . (H.) 

 For the crown = 0, and A x' = A z* ; hence 



B' = B* (HI.) 



For the left end, = a and A#= ; for the right end, = a and 

 A as" = 0; that is, 



$H r*(a 8 sin a COS a + 2 a cos* a) -f iVr 8 (3sin*a 2 a sin a COS a) 



K r* (H cos a + V sin a) a cos a A r (1 cos a) + B = 0, and 



+ ^H r 8 (a 3 sin a cos a + 2 a cos* a) i V r* (3 sin 2 a 2a sin a cos a) 

 + K 1* (H cos a + V sin a) a COS a A" r (1 COS a) + B' = 0. 



The subtraction and addition of these equation gives, after reduction, 



H r z (a 3 sin a COS a + 2 a COS 2 a) 

 ^Pr* (3 sin* a 2 a sin a cos a 2 sin*+ 2cos/3-f- 2/3 sin /3) 



+ K r 2 I 2 H a cos 2 a + P (a sin a cos a /3 sin )3) I 



+ (A-A')(l-COSa) = (TV.) 



and !Fr s sin(3sina 2acosa) 



(A + A*) r (1 cos a) K Pr'a cosa sin/3 + B + B' = . . (V.) 



