It's not circular. It rearranges into a standard quadratic equation: x^2 − 24Sx + 24 = 0. α is derived as the root of this equation.
α = (24 ± √(576 + 96S))/(48S) α = (24 + √(576 + 96·137.036...))/(48·137.036...) α = (24 + √13,723.66...)/(6577.74...) α = (24 + 117.12...)/(6577.74...) α = 141.12.../6577.74... α ≈ 0.021454...
α⁻¹ ≈ 46.61 ??? That's wrong!
α⁻¹·24α = 24αS - 1
24α²S - 24α - 1 = 0
α = (24 ± √(576 + 96S))/(48S)
α = (24 + √(576 + 96·137.036...))/(48·137.036...)
α = (24 + √13,723.66...)/(6577.74...)
α = (24 + 117.12...)/(6577.74...)
α = 141.12.../6577.74...
α ≈ 0.021454...
α⁻¹ ≈ 46.61 ???
That's wrong!
The term is -(alpha / 24). You calculated -1 / (24 · alpha).
The correct derivation is:
1 / alpha = S - (alpha / 24)
1 = S · alpha - (alpha^2) / 24
alpha^2 - 24 · S · alpha + 24 = 0
Try harder.
you have an integral and order^ power.