I have recently read a paper about the ruling span for electrical wires and they have an approximation that looks like it can be derived with mathematical intuition only. I'd like to find a derivation of this equality..
$$ S_R=\frac{\sqrt{S_1^3+S_2^3+...+S_n^3}}{\sqrt{S_1+S_2+...+S_n}}=S_{ave}+\frac{2}{3}(S_{max}-S_{ave}) $$
Update: This approximation might be geometrically motivated, I'm not entirely sure. Here is an excerpt of the definition:
The ruling span, for any series of non-uniform spans between dead ends, (S1, S2, S3 - Sn feet) is a calculated span length for which the conductor tension best represents the average tension in the conductor in the actual line.
Also, there are constraints: $S_{max}\leq\frac{3}{2}S_{ave}$ and $S_{max}\leq\frac{5}{4}S_{R}$, where $S_R$ is the ruling span in the above main equation.