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If the zeros of a polynomial p{z) when plotted in the 3-plane are symmetric in 0: 3 = 0, the zeros of the derivative p(z) of p(z) can profitably be studied by transforming onto the w-plane, with w = zf and applying known theorems there. I t is the purpose of the present note to carry that study somewhat farther than has been previously done, in particular to consider the higher derivatives of p(z). Under the transformation w = u+iv = z = (x+iy), an arbitrary line Au+Bv+C = 0 in the w-plane corresponds to an equilateral hyperbola A(x —y)+2Bxy+C = 0 in the 3-plane with center 0 or to two perpendicular lines intersecting at 0. A half-plane in the wplane for which w = 0 is an interior or exterior point corresponds in the 3-plane respectively to the exterior or interior of an equilateral hyperbola whose center is 0 ; a half-plane for which w = 0 is a boundary point corresponds to a double sector with vertex 3 = 0 and angle 7r/2. A point z is considered to be exterior or interior to a hyperbola according as the curve at its nearest point is convex or concave toward 3. We write the given polynomial in the form

On the location of the zeros of the derivatives of a polynomial symmetric in the origin