y = ab−x2.
But it has already been shown that for x = 0, y = h, hence a = h. To determine the remaining constant, the other condition may be used, that the solid formed by rotating the pressure curve represents the total pressure on the plane. The volume of the solid is
V = ∫∞0 2πxy dx
= 2πh ∫∞0 b−x2x dx
= (πh / loge b) [ −b−x2 ]∞0
= πh / loge b.
Using the condition already stated,
2ω √ (hh1) = πh / loge b,
logε b = (π/2ω) √ (h/h1).
Putting the value of b in (2) in eq. (1), and also r for the radius of the jet at the orifice, so that ω = πr2, the equation to the pressure curve is
y = hε−1/2 √(h / h1) (x2 / r2).