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-NOTESA VARIATIONAL PROBLEM FOR THE ROOTS OF A CUBIC EQUATION* (A contribution to the theory of servomechanisms) By HANS BtOKNKR (Minden, Germany) 1. The problem. Let t denote the time, u{t) a function of time and v(t) the displacement of a servomechanism, which is to simulate u(t). As regards the error n(t) = v(t) u(t) many servomechanisms obey the equation r\" + axt\" + a2v' + ci3V = —u'" — atu" (1) with constant coefficients ax , • • • a4 . The quality of the servomechanism depends on the roots of the algebraic equation x3 + ajX2 -ja2x + a3 = 0. (2) These may be denoted by xx , x2 , x3 . We consider the quantity £ Min ( —9?^ , — , — 9fa3), where SkXi means the real part of a;, . Then £ > 0 means stability, and the greater £, the faster the servomechanism follows sudden changes of u(t). A positive £ appears if and only if the well known Hurwitz-conditions are satisfied. These are a,! > 0, a2 > 0, axa2 > a3 > 0. If a, and a2 are regarded as fixed, and a3 as variable, £ is a function of a3 . An interesting problem both from a mathematical and a technical point of view, is to find Max f. There is a simple formula for it, which does not seem to be as well known as its merits. 2. The solution. Introducing the classic notations

A variational problem for the roots of a cubic equation. (A contribution to the theory of servo-mechanisms.)