CHAP. IV.] 



SUPPLEMENT JTO CHAP. XIV. 



297 



Multiplying HI. by cos a, and adding to VI., we have [(VII.) 



2 Mo sin a H r (2 sin a sin a cos a a) + ^ P r (sin a sin /3) 2 = 0. 

 In similar manner, we have from eq. (14), Art. 2, for the vertical dis- 

 placement 

 B I A y = M 8 r* (cos <j> + <t> sin <p) I H r* (2 cos <t> + 2 sin <f> sin 2 4>) 



Pr*(4> + sincoS(f>+2 sin/3 cos<f> + 24>sinsincj>) 

 K r 3 (Mo H r Pr sin )3) <t> sin </> + Ar sin <f> + O. 

 ' = M r 2 (cos<p + </>sin<^) lHr i (2cos^) + 2 <#> sin <?> sin" #) 

 + i W (<f> + sin <?> cos <p) + K r* (M H r) <f> sin <fr + A' r sin # + O'. 

 For<f> = /3, Ay = Ay', hence 



O - O' = Pr* (/3 4- sin /3 cos /3) . . . . (VHL) 

 Finally, for 4> = a, Ay = 0. For # = a, Ay' = 0, and hence 



C + O' = 2M r ! (cos a + a sina) + H> s (2 cos a -*- 2a sin a sin* a) 



+ iPr* I a + sin a cos a 2 sin (a -- ) + 2 (a 0) sin a sin /3J 



2 K r 2 (M Hr)asina + KPr*(a /3) sin a sin /3 . . .(IX.) 

 V r (a sin a cos a) = Pr (a /3 sina cos a sin /3 cos /3+2 cos a sin/3) . . (X.) 

 1. Determination of H, V and Mo. 



(a) Vertical Reaction. 



The vertical force V is given directly by eq. X. Thus 



a 3 sin a cos a sin (3 cos /3 + 2 cos a sin 



Y p 



2 (a sin a cos a) 



an expression independent of K. 

 Transforming by means of series, we have, approximately, 



For the semi-circle 



_ _ p TT 2 /3 2 sin 3 cos ff 



Sir 

 From the exact formula (43) we have the following table : 



(43) 



(44) 

 (45) 



