140. A light ray falls on a flat mirror at an angle $\alpha$. The mirror is rotated by an angle $\beta$ around the projection of the ray on the mirror. By what angle will the reflected ray deviate?
140. Let $A$ be some point on the ray, $B$ be the point of incidence of the ray on the mirror, $K$ and $L$ be the projections of $A$ onto the given mirror and the reflected one, $A_{1}$ and $A_{2}$ be the points symmetric to $A$ relative to these mirrors, respectively. The required angle is equal to the angle $\widehat{A_{1} B A_{2}}$. If $|A B|=a$, then $\left|A_{1} B\right|=\left|A_{2} B\right|=a,|A K|=a \sin \alpha$. Since $\overline{K A L}=\beta$, then $|K L|=|A K| \sin \beta=a \sin \alpha \sin \beta,\left|A_{1} A_{2}\right|=$ $=2|K L|=2 a \sin \alpha \sin \beta$. Thus, if $\varphi$ is the required angle, then ${ }^{\circ} \sin \frac{\varphi}{2}=\sin \alpha \sin \beta$.

Answer: $2 \arcsin (\sin \alpha \sin \beta)$.