Liberalia), across the road bridge, they would join the Hayling Island side of the line. This section of the line passes down the west side of the island, passing through where North Hayling station used to stand, and terminates at West Town, the main area of population in the south of the island. Sadly, due to an arson attack, the old Crossing keepers' cottages near Langston railway station unexpectedly caught fire in December 2018 and were almost completely destroyed. The property company owning the site has been ordered to rebuild them. This section is now a combined footpath, bridleway and cycleway	Hayling Island branch line
23621493	. It has recently become part of route 2 of the National Cycle Network, sponsored by Sustrans, a charity for sustainable transport. Hayling Island branch line The Hayling Island branch was a railway in Hampshire, England, that connected Havant with Hayling Island. It was sometimes known as the Hayling Billy, which is now the name of a footpath along the old track which is a Local Nature Reserve. History. The line was opened by the London, Brighton and South Coast Railway (LBSCR) for goods on 19 January 1865, and for passengers on 16 July 1867. It ran from Havant to Hayling 	Hayling Island branch line
23621494	Weierstrass–Enneper parameterization Weierstrass–Enneper parameterization In mathematics, the Weierstrass–Enneper parameterization of minimal surfaces is a classical piece of differential geometry. Alfred Enneper and Karl Weierstrass studied minimal surfaces as far back as 1863. Let formula_1 and formula_2 be functions on either the entire complex plane or the unit disk, where formula_2 is meromorphic and formula_1 is analytic, such that wherever formula_2 has a pole of order formula_6, formula_1 has a zero of order formula_8 (or equivalently, such that the product formula_9 is holomorphic), and let formula_10 be constants. Then the surface with coordinates formula_11 is minimal, where the formula_12 are defined 	Weierstrass–Enneper parameterization
23621495	using the real part of a complex integral, as follows: formula_13 The converse is also true: every nonplanar minimal surface defined over a simply connected domain can be given a parametrization of this type. For example, Enneper's surface has , . Parametric surface of complex variables. The Weierstrass-Enneper model defines a minimal surface formula_14 (formula_15) on a complex plane (formula_16). Let formula_17 (the complex plane as the formula_18 space), the Jacobian matrix of the surface can be written as a column of complex entries: formula_19 qhere formula_20 and formula_21 are holomorphic functions of formula_22. The Jacobian formula_23 represents the two 	Weierstrass–Enneper parameterization
23621496	orthogonal tangent vectors of the surface: formula_24 The surface normal is given by formula_25 The Jacobian formula_23 leads to a number of important properties: formula_27, formula_28, formula_29, formula_30. The proofs can be found in Sharma's essay: The Weierstrass representation always gives a minimal surface. The derivatives can be used to construct the first fundamental form matrix: formula_31 and the second fundamental form matrix formula_32 Finally, a point formula_33 on the complex plane maps to a point formula_34 on the minimal surface in formula_35 by formula_36 where formula_37 for all minimal surfaces throughout this paper except for Costa's minimal 	Weierstrass–Enneper parameterization
23621497	surface where formula_38. Embedded minimal surfaces and examples. The classical examples of embedded complete minimal surfaces in formula_39 with finite topology include the plane, the catenoid, the helicoid, and the Costa's minimal surface. Costa's surface involves Weierstrass's elliptic function formula_40: formula_41 formula_42 where formula_43 is a constant. Helicatenoid. Choosing the functions formula_44 and formula_45, a one parameter family of minimal surfaces is obtained. formula_46 formula_47 formula_48 formula_49 Choosing the parameters of the surface as formula_50: formula_51 At the extremes, the surface is a catenoid formula_52 or a helicoid formula_53. Otherwise, formula_54 represents a mixing angle. The resulting 	Weierstrass–Enneper parameterization
23621498	surface, with domain chosen to prevent self-intersection, is a catenary rotated around the formula_55 axis in a helical fashion. Lines of curvature. One can rewrite each element of second fundamental matrix as a function of formula_1 and formula_2, for example formula_58 And consequently the second fundamental form matrix can be simplified as formula_59 One of its eigenvectors is formula_60 which represents the principal direction in the complex domain. Therefore, the two principal directions in the formula_18 space turn out to be formula_62 Weierstrass–Enneper parameterization In mathematics, the Weierstrass–Enneper parameterization of minimal surfaces is a classical piece of 	Weierstrass–Enneper parameterization
23621499	Beautiful People Beautiful People Beautiful People may refer to: Beautiful People Beautiful People may refer to: 	Beautiful People
23621500	Liberalia Liberalia The Liberalia (March 17) is the festival of Liber Pater and his consort Libera. The Romans celebrated Liberalia with sacrifices, processions, ribald and gauche songs, and masks which were hung on trees. This feast celebrates the maturation of young boys to manhood. Roman boys, usually at age 15 or 16, would remove the bulla praetexta", a hollow charm of gold or leather, which parents placed about the necks of children to ward off evil spirits. At the Liberalia ceremony the young men might place the bulla on an altar (with a lock of hair or the stubble of his  first shave placed inside) and dedicate it to the Lares, who were gods of the household and family. Mothers often retrieved the discarded bulla praetexta and kept it out of superstition. If the son ever achieved a public triumph, the mother could display the bulla to ward off any evil that might be wished upon the son by envious people. The young men discarded the toga praetexta, which was probably derived from Etruscan dress and was decorated with a broad purple border and worn with the bulla, by boys and girls. The boys donned the clothing of adulthood, the pure  white toga virilis, or "man's gown". The garment identified him as a citizen of Rome, and thus an eligible voter. The celebration on March 17 was meant to honor Liber Pater, an ancient god of fertility and wine (like Bacchus, the Roman version of the Greek god Dionysus). Liber Pater is also a vegetation god, responsible for protecting seed. Liber, again like Dionysus, had female priests although Liber's priests were older women, known as Sacerdos Liberi. Wearing wreaths of ivy, the priestesses made special cakes, or libia, of oil and honey which passing devotees would have them sacrifice  on their behalf. Over time this feast evolved and included the goddess Libera, Liber Pater's consort, and the feast divided so that Liber governed the male seed and Libera the female. Ovid in his almanac entry for the festival identifies Libera as the celestial manifestation of Ariadne. This ancient Italian ceremony was a "country" or rustic ceremony. The processional featured a large phallus which the devotees carried throughout the countryside to bring the blessing of fertility to the land and the people. The procession and the phallus were meant also to protect the crops from evil. At the end  of the procession, a virtuous and respected matron placed a wreath upon the phallus. Related to the celebration of the Liberalia is the Procession of the Argei, celebrated on March 16 and 17. The Argei were 27 sacred shrines created by the Numina (very powerful ancient gods who are divine beings without form or face) and found throughout the regions of Rome. However, modern scholars have not discovered their meaning or use. In the argei celebration, 30 figures also called Argei were fashioned from rushes into shapes resembling men; later in the year they were tossed into the river(s). The  origin of this celebration is not certain, but many scholars feel that it may have been a ritualistic offering meant to appease and praise the Numa and that the 30 argei probably represented the thirty elder Roman curiae, or possibly represented the 30 Latin townships. Other ancient scholars wrote that the use of the bull-rush icons was meant to deter celebrants from human sacrifice, which was done to honor Saturn. Some historical documents indicate that the argei (the sacred places) took their names from the chieftains who came with Hercules, the Argive, to Rome and then occupied the Capitoline  (Saturnian) Hill. There is no way at present to verify this information, but it does coincide with the belief that Rome was founded by the Pelasgians and the name Argos is linked to that group. While Liberalia is a relatively unknown event in modern times, references to Liberalia and the Roman goddess Libera are still found today online and in astrology. Liberalia The Liberalia (March 17) is the festival of Liber Pater and his consort Libera. The Romans celebrated Liberalia with sacrifices, processions, ribald and gauche songs, and masks which were hung on trees. This feast celebrates the maturation of