x (1 + λ/a2) = 0, y (1 + λ/b2) = 0, z (1 + λ/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, λ = −a2; (2) z = 0, x = 0, λ = −b2; |
| (3) x = 0, y = 0, λ = −c2. |
In the case of (1) δu = δy2 (1 − a2/b2) + δz2 (1 − a2/c2), which is always negative, so that u = a2 gives a maximum.
In the case of (3) δu = δx2 (1 − c2/a2) + δy2 (1 − c2/b2), which is always positive, so that u = c2 gives a minimum.
In the case of (2) δu = δx2 (1 − b2/a2) − δz2(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.
Editions by J. C. Wernsdorf, Poetae latini minores, vi.; E. Bährens, Poetae latini minores, v.; M. Petschenig (1890), in C. F. Ascherson’s Berliner Studien, xi.; R. Webster (Princeton, 1901; see Classical Review, Oct. 1901), with introduction and commentary; see also Robinson Ellis in American Journal of Philology, v. (1884) and Teuffel-Schwabe, Hist. of Roman Literature (Eng. trans.), § 490. There is an English version (as from Cornelius Gallus), by Hovenden Walker (1689), under the title of The Impotent Lover.