A steel bridge expands at a rate of $0.850 \text{ inches per } 100^\circ\text{F}$ of temperature increase. Express this expansion rate in millimeters per $100^\circ\text{C}$ of temperature increase. (Take $1 \text{ inch} = 25.4 \text{ mm}$ and a temperature change of $1 ^\circ\text{C} = 1.8 ^\circ\text{F}$)