CHAP. H.] SUPPLEMENT TO CHAP. XIV. 283 



and accordingly A - A' = (V - V) a z - \ (V + V) a 2 . Since V + V = P, 

 and from (16) V V = P , we have, 



Of 



A-A' = iPa 2 ....... (I.) 



(bj Horizontal displacement. 



Inserting the value of dy for the parabola, viz., ay = - xdx, and the 



a 



value of M from (19), and inserting in this last the value of y, viz., 



y = as 2 , we have from equation (15), Art. 2, 

 a 1 



Integrating this, we have for the three divisions of the arch, as before, 



B "I 

 J 







5aV \ 8 a 



- \ - t V'ose A +5 

 a 5aV \ 8a 



For the point B, or a? = 3, we have from the two first of these equations, 

 B - B' = ^ (V - V) a z 3 - i (V + V) 3* - i (A - A') a 2 , that is, 



B-B' = -,frP* ....... (EL) 



For the crown, a? = 0, and the second and third equations are equal, 

 hence 



B' = B" . ....... (HI.) 



For the left end, that is, for x = a, since the end of the arch must not slip, 

 we must have x' * = 0. So also for the right end, for x = a. There- 

 fore, from the first and third equations, putting B' for B", we have 



& H a 3 h - ^ V a 4 + \ A a 2 + B = 0, 

 - f s H a 3 h + -ff V a* + i A" a 2 + B' = 0. 



By the addition and subtraction of these equations, we have, since 

 V + V = P, and (V - V) a = Pz, 



A')a 2 -(B + B') = .... (IV.) 



+ a 4 ) + i(A-A')a 2 =0 . . (V.) 



We might, in a precisely similar manner, form three equations similar to 

 (23) for the vertical displacement A y. This would introduce three more 

 constants and four more equations of condition between them. By the 

 nine equations I. to IX. thus obtained, these constants may be then deter- 

 mined in terms of the known quantities H, A, P, a and a, and thus the 

 change of shape at any point may be fully determined. 

 The complete discussion, as indicated, is unnecessary for the purpose, we 



