The commutative law in arithmetic, for instance, states that **adb** and b+a, or ab and ba, are equal.

If the ray suffers one internal reflection at D, then it is readily seen that, if DB be the path of the reflected ray, the angle **ADB** equals 2r, i.e.

7 angled at C, **ADB** is the semicircle described on AB as diameter, AEB the circular arc described with centre C and radius CA= CB.

The arm CD turns on the axis C, and -.~ .- is jointed at D to the middle of the bar **ADB**, whose length is double I of that of CD, and one of whose - ends B is jointed to a slider, sliding in straight guides along the line FIG 113.

Draw BE perpendicular to CB, cutting CD produced in E, then E is the instantaneous axis of the bar **ADB**; and the direction of motion of A is at every instant perpendicular to EAthat is, along the straight line ACa.