Stationary deviation of a primary rainbow ray ($\cos \alpha =(n/2)\cos \beta$)

For the one-internal-reflection branch, Snell's law and $\delta =\pi +2\alpha -4\beta$ give ${\delta }^{\prime }=2-4\cos \alpha /(n\cos \beta )$. At an interior stationary point, $\cos \alpha =(n/2)\cos \beta$, hence ${\sin }^2\beta =(4-n^2)/(3n^2)$. Another differentiation gives ${\delta }^{\prime \prime }=3\tan \beta >0$ there. The interior minimum exists for $1<n<2$. The internal specular reflection can be partial; total internal reflection is not required.