and the rest of the solution proceeds as before in § 14, putting
A half wave length of the curve of gyrostats is covered when ck = π, so that π/c is the number of gyrostats in a half wave, which is therefore of wave length 2π (a + l)/c.
A plane polarized wave is given when exp cki is replaced by exp (nt + ck)i, and a wave circularly polarized when w, ῶ, σ of § 14 replace this x, θ, α.
Gyroscopic Pendulum.—The elastic flexure joint is useful for supporting a rod, carrying a fly-wheel, like a gyroscopic pendulum.
Expressed by Euler’s angles, θ, φ, ψ, the kinetic energy is
where A refers to rod and gyroscope about the transverse axis at the point of support, C′ refers to rod about its axis of length, and C refers to the revolving fly-wheel.
The elimination of ψ. between the equation of conservation of angular momentum about the vertical, viz.
(13) A sin2 θψ. − C′ (1 − cos θ) cos θψ. + C(φ. + ψ. cos θ) cos θ = G, a constant, and the equation of energy, viz.
(14) T − gMh cos θ = H, a constant, with θ measured from the downward vertical, and
(15) φ. + ψ. cos θ = R, a constant, will lead to an equation for dθ/dt, or dz/dt, in terms of cos θ or z, the integral of which is of hyperelliptic character, except when A = C′.
In the suspension of fig. 8, the motion given by φ. is suppressed in the stalk, and for the fly-wheel φ. gives the rubbing angular velocity of the wheel on the stalk; the equations are now
and the motion is again of hyperelliptic character, except when A = C′, or C′＝0. To realize a motion given completely by the elliptic function, the suspension of the stalk must be made by a smooth ball and socket, or else a Hooke universal joint.
Finally, there is the case of the general motion of a top with a spherical rounded point on a smooth plane, in which the centre of gravity may be supposed to rise and fall in a vertical line. Here
with θ measured from the upward vertical, and
where A now refers to a transverse axis through the centre of gravity.
The elimination of ψ. leads to an equation for z, = cos θ, of the form
| (20) | ( | dz | ) | 2 | = 2 | g | Z | = 2 | g | (z1 − z) (z2 − z) (z3 − z) | , | |||
| dt | h | 1 − z2 + A/Mh2 | h | (z4 − z) (z − z5) |
with the arrangement
so that the motion is hyperelliptic.
Authorities.—In addition to the references in the text the following will be found useful:—Ast. Notices, vol. i.; Comptes rendus, Sept. 1852; Paper by Professor Magnus translated in Taylor’s Foreign Scientific Memoirs, n.s., pt. 3, p. 210; Ast. Notices, xiii. 221-248; Theory of Foucault’s Gyroscope Experiments, by the Rev. Baden Powell, F.R.S.; Ast. Notices, vol. xv.; articles by Major J. G. Barnard in Silliman’s Journal, 2nd ser., vols. xxiv. and xxv.; E. Hunt on “Rotatory Motion,” Proc. Phil. Soc. Glasgow, vol. iv.; J. Clerk Maxwell, “On a Dynamical Top,” Trans. R.S.E. vol. xxi.; Phil. Mag. 4th ser. vols. 7, 13, 14; Proc. Royal Irish Academy, vol. viii.; Sir William Thomson on “Gyrostat,” Nature, xv. 297; G. T. Walker, “The Motion of a Celt,” Quar. Jour. Math., 1896; G. T. Walker, Math. Ency. iv. 1, xi. 1; Gallop, Proc. Camb. Phil. Soc. xii. 82, pt. 2, 1903, “Rise of a Top”; Price’s Infinitesimal Calculus, vol. iv.; Worms, The Earth and its Mechanism; Routh, Rigid Dynamics; A. G. Webster, Dynamics (1904); H. Crabtree, Spinning Tops and Gyroscopic Motion (1909). For a complete list of the mathematical works on the subject of the Gyroscope and Gyrostat from the outset, Professor Cayley’s Report to the British Association (1862) on the Progress of Dynamics should be consulted. Modern authors will be found cited in Klein and Sommerfeld, Theorie des Kreisels (1897), and in the Encyclopädie der mathematischen Wissenschaften. (G. G.)
GYTHIUM, the harbour and arsenal of Sparta, from which it
was some 30 m. distant. The town lay at the N.W. extremity of
the Laconian Gulf, in a small but fertile plain at the mouth of the
Gythius. Its reputed founders were Heracles and Apollo, who
frequently appear on its coins: the former of these names may
point to the influence of Phoenician traders, who, we know,
visited the Laconian shores at a very early period. In classical
times it was a community of perioeci, politically dependent on
Sparta, though doubtless with a municipal life of its own. In
455 B.C., during the first Peloponnesian War, it was burned
by the Athenian admiral Tolmides. In 370 B.C. Epaminondas
besieged it unsuccessfully for three days. Its fortifications were
strengthened by the tyrant Nabis, but in 195 B.C. it was invested
and taken by Titus and Lucius Quintius Flamininus, and,
though recovered by Nabis two or three years later, was recaptured
immediately after his murder (192 B.C.) by Philopoemen
and Aulus Atilius and remained in the Achaean League until its
dissolution in 146 B.C. Subsequently it formed the most important
of the Eleutherolaconian towns, a group of twenty-four,
later eighteen, communities leagued together to maintain their
autonomy against Sparta and declared free by Augustus. The
highest officer of the confederacy was the general (στρατηγός),
who was assisted by a treasurer (ταμίας), while the chief
magistrates of the several communities bore the title of ephors
(ἔφοροι).
Pausanias (iii. 21 f.) has left us a description of the town as it existed in the reign of Marcus Aurelius, the agora, the Acropolis, the island of Cranae (Marathonisi) where Paris celebrated his nuptials with Helen, the Migonium or precinct of Aphrodite Migonitis (occupied by the modern town of Marathonisi or Gythium), and the hill Larysium (Koumaro) rising above it. The numerous remains extant, of which the theatre and the buildings partially submerged by the sea are the most noteworthy, all belong to the Roman period.
The modern town is a busy and flourishing port with a good harbour protected by Cranae, now connected by a mole with the mainland: it is the capital of the prefecture (νομός) of Λακωνική with a population in 1907 of 61,522.
See G. Weber, De Gytheo et Lacedaemoniorum rebus navalibus (Heidelberg, 1833); W. M. Leake, Travels in the Morea, i. 244 foll.; E. Curtius, Peloponnesos, ii. 267 foll. Inscriptions: Le Bas-Foucart, Voyage archéologique, ii. Nos. 238-248 f.; Collitz-Bechtel, Sammlung d. griech. Dialekt-Inschriften, iii. Nos. 4562-4573; British School Annual, x. 179 foll. Excavations: Ἀ. Σκιᾶς, Πρακτικὰ τῆς Ἀρχ. Ἑταιρείας, 1891, 69 foll. (M. N. T.)
GYULA-FEHÉRVÁR (Ger. Karlsburg), a town of Hungary, in
Transylvania, in the county of Alsó-Feliér, 73 m. S. of Kolozsvár
by rail. Pop. (1900) 11,507. It is situated on the right bank of
the Maros, on the outskirts of the Transylvanian Erzgebirge or
Ore Mountains, and consists of the upper town, or citadel, and
the lower town. Gyula-Fehérvár is the seat of a Roman Catholic
bishop, and has a fine Roman Catholic cathedral, built in the
11th century in Romanesque style, and rebuilt in 1443 by
John Hunyady in Gothic style. It contains among other tombs
that of John Hunyady. Near the cathedral is the episcopal
palace, and in the same part of the town is the Batthyaneum,
founded by Bishop Count Batthyány in 1794. It contains a
valuable library with many incunabula and old manuscripts,
amongst which is one of the Nibelungenlied, an astronomical
observatory, a collection of antiquities, and a mineral collection.
Gyula-Fehérvár carries on an active trade in cereals, wine and
cattle.
Gyula-Fehérvár occupies the site of the Roman colony Apulum. Many Roman relics found here, and in the vicinity, are preserved in the museum of the town. The bishopric was founded in the 11th century by King Ladislaus I. (1078–1095). In the 16th century, when Transylvania separated from Hungary, the town became the residence of the Transylvanian princes. From this period dates the castle, and also the buildings of the university, founded by Gabriel Bethlen, and now used as barracks. After the reversion of Transylvania in 1713 to the Habsburg monarchy the actual strong fortress was built in 1716–1735 by the emperor Charles VI., whence the German name of the town.
