CHAP. IV.] SUPPLEMENT TO CHAP. XIV. 295 



From (34) and (35) we can now find the intersection and direction curves, 

 The preceding table gives us sufficient data for complete calculation by 

 moments according to Art. 162. The intersection and direction curves 

 will, as already explained, enable us to find the above quantities graphi- 

 cally. 



15. Intersection Curve. From (33) we have y = - ' , or in- 



XI 



serting the values of H, V and M above, and reducing, 



For the parabolic arch with fixed ends, then, the intersection curve become* 

 a straight horizontal line, i h above the crown. 



16. Direction Curve. From (36), (37) and (38) we have 



dz ~ 8a*h ' da 4 a 1 ' 



d Mo _ P (a - z) (4a* - 5 az - 5 s>) 

 ds ' S a* 



Inserting these in (35), as also the values of H, V and M themselves, and 

 reducing, we have 



..... (39) 



For a = 0, t> = fa, w = $h. For s a, = 



For z= , v = a, w = GO . Eliminating z, we have 



5 2 Sae + Sc* 

 > = - 77-^ - r - * ...... (40) 



15 a (a c) 



This is the equation of an hyperbola. Hence, for the parabolic arc with 

 fixed ends, the direction curve is upon each side of the crown an hyperbola, 

 This hyperbola is described in Art. 160 of the text, (Fig. 93), and a table 

 to facilitate its construction is there given. 



B. CIRCULAR ARC CONSTANT CROSS-SECTION CONCENTRATED IX)AD. 



17. Fundamental Equations. From eq. (32) we have, since 



x r sin <, y = r (1 cos <j>), z = r sin ft 



M = M Hr(l cos$) + (P V)rsin0 Prsin | 

 M' = M. -Hr(l-co3<)- Vrsin0 j ' ( 



The expressions for G, Art. 13, eq. (31), apply here directly. 

 Therefore, from eq. (8), Art 1, we have 



