The equation is the bipolar equation of the Cayley's oval. Arthur Cayley (1821-1895) studied the curve in 1857.
And together in one field:
1) While the force is of the form F = k/r2, the potential has the form V=k/r.
2) Realize that r12 = (x+1)2 + y2, r22 = (x-1)2 + y2