We take the locus of the lemniscate, and at each point of the locus construct a normal to the curve. The evolute occurs as the envelope of these normals.

In the applet linked below, as Q traverses the red circle, point P describes the lemniscate, and the normal at P contributes to the envelope of the evolute. Press the ‘Animate’ button to generate the evolute. You can delete this construction by clicking on the red cross at lower right. A fuller representation of the locus can be found by manually dragging the point Q around the red circle. It is difficult to see exactly what is happening here. Check the behaviour when points P and Q are close together.