for all values of r. These equations with the equations u1 = 0, ..., u_m = 0 are exactly enough to determine [lambda]1, ..., [lambda]_m, x1 x2, ..., x_n, so that we find critical values of u, and examine the terms of the second order to decide whether we obtain a maximum or minimum. To take a very simple illustration; consider the problem of determining the maximum and minimum radii vectors of the ellipsoid x²/a² + y²/b² + z²/c² = 1, where a² > b² > c². Here we require the maximum and minimum values of x² + y² + z² where x²/a² + y²/b² + z²/c² = 1. We have / [lambda]\ / [lambda]\ / [lambda]\ [delta]u = 2x [delta]x ( 1 + -------- ) + 2y [delta]y ( 1 + -------- ) + 2z [delta]z ( 1 + -------- ) \ a² / \ b² / \ c² / / [lambda]\ / [lambda]\ / [lambda]\ + [delta]x² ( 1 + -------- ) + [delta]y² ( 1 + -------- ) + [delta]z² ( 1 + -------- ). \ a² / \ b² / \ c² / To make the terms of the first order disappear, we have the three equations:-- x(1 + [lambda]/a²) = 0, y(1 + [lambda]/b²) = 0, z(1 + [lambda]/c²) = 0. These have three sets of solutions consistent with the conditions x²/a² + y²/b² + z²/c² = 1, a² > b² > c², viz.:-- (1) y = 0, z = 0, [lambda] = -a²; (2) z = 0, x = 0, [lambda] = -b²; (3) x = 0, y = 0, [lambda] = -c². In the case of (1) [delta]u = [delta]y² (1 - a²/b²) + [delta]z² (1 - a²/c²), which is always negative, so that u = a² gives a maximum. In the case of (3) [delta]u = [delta]x² (1 - c²/a²) + [delta]y² (1 - c²/b²), which is always positive, so that u = c² gives a minimum. In the case of (2) [delta]u = [delta]x²(1 - b²/a²) - [delta]z²(b²/c² - 1), which can be made either positive or negative, or even zero if we move in the planes x²(1 - b²/a²) = z²(b²/c² - 1), which are well known to be the central planes of circular section. So that u = b², though a critical value, is neither a maximum nor minimum, and the central planes of circular section divide the ellipsoid into four portions in two of which a² > r² > b², and in the other two b² > r² > c². (A. E. J.) MAXIMIANUS, a Latin elegiac poet who flourished during the 6th century A.D. He was an Etruscan by birth, and spent his youth at Rome, where he enjoyed a great reputation as an orator. At an advanced age he was sent on an important mission to the East, perhaps by Theodoric, if he is the Maximianus to whom that monarch addressed a letter preserved in Cassiodorus (_Variarum_, i. 21). The six elegies extant under his name, written in old age, in which he laments the loss of his youth, contain descriptions of various amours. They show the author's familiarity with the best writers of the Augustan age.