−GQ dz − Q dp − ζG (χ/Ω) Q (v2/2g) dl = 0.
Dividing by GQ,
dz + dp/G + ζ (χ/Ω) (v2/2g) dl = 0.
Integrating,
z + p/G + ζ (χ/Ω) (v2/2g) l = constant.
(1)
§ 72. Let A and B (fig. 81) be any two sections of the pipe for which p, z, l have the values p1, z1, l1, and p2, z2, l2, respectively. Then
z1 + p1/G + ζ (χ/Ω) (v2/2g) l1 = z2 + p2/G + ζ (χ/Ω) (v2/2g) l2;
or, if l2 − l1 = L, rearranging the terms,
ζv2/2g = (1/L) {(z1 + p1/G) − (z2 + p2/G)} Ω/χ.