 Solution

Use unambiguous symbols ${\gamma }_{SV},{\gamma }_{SL},{\gamma }_{LV}$ for solid–vapor, solid–liquid and liquid–vapor interfacial energies. The printed description repeats an interface subscript, so it cannot be used literally for two different interfaces.

For a smooth homogeneous equilibrium surface, take a horizontal displacement of the contact line. The substrate-parallel force balance isThus the contact-angle relation is Young's equation:The same result follows by varying interfacial free energy at fixed liquid volume. It assumes equilibrium, no contact-line pinning, and negligible line-tension and roughness corrections. If the inferred cosine lies outside $[-1,1]$, the finite-angle equilibrium construction fails and a wetting limit must be considered.

The name in the question should be distinguished from the Young–Laplace equation, which relates a pressure jump to interface curvature. The horizontal interfacial balance derived here is Young's equation; it does not require introducing a pressure-curvature formula.