= Solution
At a straight <contact line>, take forces per unit transverse length and measure the <contact angle> $\theta$ through the liquid. Solid–vapor <surface tension> pulls outward along the solid, solid–liquid <surface tension> pulls inward, and the liquid–vapor force has horizontal component $\gamma_{LV}\cos\theta$. Tangential equilibrium gives
$$
\gamma_{SV}-\gamma_{SL}-\gamma_{LV}\cos\theta=0,\qquad
\boxed{\cos\theta=\frac{\gamma_{SV}-\gamma_{SL}}{\gamma_{LV}}}.
$$
This is <Young's equation>. The substrate supplies the normal reaction balancing the vertical capillary component. The original force sketch labels directions at a right-hand contact line. The result assumes smooth, chemically uniform, rigid surfaces in partial-wetting equilibrium, with negligible pinning and line tension.
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