296 SUPPLEMENT TO CHAP. XTV. [CHAP. IV 



Hence from (12), Art. 1, since di = rd(f>, 



' = =-=-1 Mo Hr (1 cos <) V> sin < I d(f)-\ 



d A < = lUc-Hr (1 cos0) + (P-V) r sin 0-Pr sin 3 d<}>- d p, 



'-. 0. 



Substituting the values of above, integrating, and putting, as bo- 

 as 



fore, for brevity, K = , we have 

 Ar 8 



o< Hr($ sin 0) (P V)r cos $ 

 + K r (M Hr - P r sin ) + A. 

 $- Hr(# sin <) + Vrcos^J + cr (M 

 For < = |3, A <f> = A #', and we obtain 



A- A' = Pr*[cos/3 + (1 +K)j3sinj3J ... (I.) 



For <f> = a, A <p = 0, and f or < = a, A $>' = 0. Adding and subtracting 

 the equations thus obtained, and eliminating A A', we have 



A + A' = P r* [cos a + (! + *) sin 0J - SVr'cosa . . (IL) 

 2 Mo a 2 H r (a sin a) Pr I cos a cos + (a /3) sin /3J 



+ K [2 Mo a 2H r a - P r (a /3) sin /3J = . . (HI.) 



From eq. (14), Art. 2, we have, as before, after integrating, for the hori- 

 zontal displacement, 



E I Aa:= Mo r 2 (sin <p <t> cos <?>)+ J Hr 8 (2 sin $> 2 <f> cos <#> ^ + sin cos $>) 

 i Vr 3 sin a ^ + P r 8 (sin 2 <^ + 2 sin /3 sin <f> 2 $ sin /3 cos <f>) 



+ K r" (M Hr P r sin ) <^ cos + A r cos <f> + B. 

 El Aa?"= Mor s (sin<f <^ cos </>)+Hr'(2sin ^> 2^> cos ^ </>+ sin^cos ^>) 



i V r* sin* # + K r 9 (M H r) # cos ^> + A' r cos <f> + B'. 

 For <f> = /5, A a? = A a;', hence 



B - B' = - i P r 3 (2 + sin* ft) . . . . (IV.) 



Further, for <f> = a, A a: = 0, and for <p = a, A a?' = 0. Hence, by add- 

 ing and subtracting, 



B + B' = V r 8 (2 sin 2 o) i Pr ! (2 - sin* a + 2 sin a sin 0). .(V.) 

 2 Mo r* (sin a a cos a) H r 3 (2 sin a 2 a cos a a + sin a cos a) 



+ | P r 8 [~2 sin 4 o + Bin 3 2 cos (o j8) -h 2 (a ) cos a sin j8 J 

 - <c r 2 ["2 (Mo - Hr) a cos a - Pr (o - 0) COB a sin 0J = 0. .(VI). 



