|Σ||δu||δx1 + λ1 Σ||δu1||δx1 + ... + λm Σ||δum||δx1.|
We can choose λ1, ... λm, to make the coefficients of δx1, δx2, ... δxm, vanish, and the remaining δxm+1 to δxn may be regarded as independent, so that, when u has a critical value, their coefficients must also vanish. So that we put
|δu||+||δu1||+ ... + λm||δum||= 0|
for all values of r. These equations with the equations u1 = 0, ..., um = 0 are exactly enough to determine λ1, ..., λm, x1 x2, ..., xn, 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.
|δu = 2xδx ( 1 +||λ||) + 2yδy (||λ||) + 2zδz (||λ||)|
|+ δx2 ( 1 +||λ||) + δy2 (||λ||) + δz2 (||λ||).|
To make the terms of the first order disappear, we have the three equations:—
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.
MAXIMIANUS, MARCUS AURELIUS VALERIUS, surnamed Herculius, Roman emperor from A.D. 286 to 305, was born of humble parents at Sirmium in Pannonia. He achieved distinction during long service in the army, and having been made Caesar by Diocletian in 285, received the title of Augustus in the following year (April 1, 286). In 287 he suppressed the rising of the peasants (Bagaudae) in Gaul, but in 289, after a three years’ struggle, his colleague and he were compelled to acquiesce in the assumption by his lieutenant Carausius (who had crossed over to Britain) of the title of Augustus. After 293 Maximianus left the care of the Rhine frontier to Constantius Chlorus, who had been designated Caesar in that year, but in 297 his arms achieved a rapid and decisive victory over the barbarians of Mauretania, and in 302 he shared at Rome the triumph of Diocletian, the last pageant of the kind ever witnessed by that city. On the 1st of May 305, the day of Diocletian’s abdication, he also, but without his colleague’s sincerity, divested himself of the imperial dignity at Mediolanum (Milan), which had been his capital, and retired to a villa in Lucania; in the following year, however, he was induced by his son Maxentius to reassume the purple. In 307 he brought the emperor Flavius Valerius Severus a captive to Rome, and also compelled Galerius to retreat, but in 308 he was himself driven by Maxentius from Italy into Illyricum, whence again he was compelled to seek refuge at Arelate (Arles), the court of his son-in-law, Constantine. Here a false report was received, or invented, of the death of Constantine, at that time absent on the Rhine. Maximianus at once grasped at the succession, but was soon driven to Massilia (Marseilles), where, having been delivered up to his pursuers, he strangled himself.
See Zosimus ii. 7-11; Zonaras xii. 31-33; Eutropius ix. 20, x. 2, 3; Aurelius Victor p. 39. For the emperor Galerius Valerius Maximianus see Galerius.
MAXIMILIAN I. (1573–1651), called “the Great,” elector and duke of Bavaria, eldest son of William V. of Bavaria, was born at Munich on the 17th of April 1573. He was educated by the Jesuits at the university of Ingolstadt, and began to take part in the government in 1591. He married in 1595 his cousin, Elizabeth, daughter of Charles II., duke of Lorraine, and became duke of Bavaria upon his father’s abdication in 1597. He refrained from any interference in German politics until 1607, when he was entrusted with the duty of executing the imperial ban against the free city of Donauwörth, a Protestant stronghold. In December 1607 his troops occupied the city, and vigorous steps were taken to restore the supremacy of the older faith. Some Protestant princes, alarmed at this action, formed a union to defend their interests, which was answered in 1609 by the establishment of a league, in the formation of which Maximilian took an important part. Under his leadership an army was set on foot, but his policy was strictly defensive and he refused to allow the league to become a tool in the hands of the house of Habsburg. Dissensions among his colleagues led the duke to resign his office in 1616, but the approach of trouble brought about his return to the league about two years later.
Having refused to become a candidate for the imperial throne in 1619, Maximilian was faced with the complications arising from the outbreak of war in Bohemia. After some delay he made a treaty with the emperor Ferdinand II. in October 1619, and in return for large concessions placed the forces of the league at the emperor’s service. Anxious to curtail the area of the struggle, he made a treaty of neutrality with the Protestant Union, and occupied Upper Austria as security for the expenses of the campaign. On the 8th of November 1620 his troops under Count Tilly defeated the forces of Frederick, king of Bohemia and count palatine of the Rhine, at the White Hill near Prague. In spite of the arrangement with the union Tilly then devastated the Rhenish Palatinate, and in February 1623 Maximilian was formally invested with the electoral dignity and the attendant office of imperial steward, which had been enjoyed since 1356 by the counts palatine of the Rhine. After receiving the Upper Palatinate and restoring Upper Austria to Ferdinand, Maximilian became leader of the party which sought to bring about Wallenstein’s dismissal from the imperial service. At the diet of Regensburg in 1630 Ferdinand was compelled to assent to this demand, but the sequel was disastrous both for Bavaria and its ruler. Early in 1632 the Swedes marched into the duchy and occupied Munich, and Maximilian could only obtain the assistance of the imperialists by placing himself under the orders of Wallenstein, now restored to the command of the emperor’s forces. The ravages of the Swedes and their French allies induced the elector to enter into negotiations for peace with Gustavus Adolphus and Cardinal Richelieu. He also proposed to disarm the Protestants by modifying the Restitution edict of 1629; but these efforts were abortive. In March 1647 he concluded an armistice with France and Sweden at Ulm, but the entreaties of the emperor Ferdinand III. led him to disregard his undertaking. Bavaria was again ravaged, and the elector’s forces defeated in May 1648 at Zusmarshausen. But the peace of Westphalia soon put an end to the struggle. By this treaty it was agreed that Maximilian should retain the electoral dignity, which was made hereditary in his family; and the Upper Palatinate was incorporated with Bavaria. The elector died at Ingolstadt on the 27th of September 1651. By his second wife,