we may write x = sn u, √(1 - x2) = cn u, √(1 - k2x2) = dn u. These are Jacobi's elliptic functions. They obviously reduce to sin u, cos u, 1, when k is 0. The circular functions have the period 2π; the elliptic functions are doubly periodic, having both a real and a pure imaginary period when k2 is real and less than 1. Like the functions sine and cosine, the elliptic functions have addition theorems, e.g.
| sn(u + v) = | sn u cn v dn v + sn v cn u dn u | . |
| 1 - k2sn2u sn2v |
Another method and notation has been introduced by Weierstrass, and is now much used. The functions are needed for the solution of many physical problems, such as those of the motion of a top and of a pendulum.—Bibliography: A. G. Greenhill, Elliptic Functions; Appell and Lacour, Fonctions Elliptiques; Whittaker and Watson, Modern Analysis.
Ellis, Alexander John, English philologist, born 1814 (name originally Sharpe), died in 1890. He was a distinguished graduate of Trinity College, Cambridge, was elected to the Royal Society in 1864, and was long a prominent member of the Philological Society, being more than once its president. Though phonetics was the subject in which he most highly distinguished himself, he was equally at home in mathematical and musical subjects. His chief published work is Early English Pronunciation (in five parts), between 1869 and 1889; but his publications in the form of books, pamphlets, papers, and articles on phonetics, music, and mathematics are numerous.
Ellis, George, English man of letters, born in 1753, died in 1815. Educated at Westminster School and Trinity College, Cambridge, he became one of the junta of wits concerned in the well-known political satire, The Rolliad, and contributed to the Anti-Jacobin. He also wrote a preface, notes, and appendix to Way's translation of Le Grand's Fabliaux, and published Specimens of the Early English Poets, with an Historical Sketch (1790), and Specimens of Early English Metrical Romances (1805). He was an intimate friend of Sir Walter Scott.
Ellis, William, English missionary, born 1794, died 1872. He was sent out to the South Sea Islands in 1816 by the London Missionary Society, and returned in 1825, one result of his labours being Polynesian Researches (1829). From 1830 to 1844 he was secretary to the society, and afterwards on its behalf made several visits to Madagascar, the longest being from 1861 to 1865. These visits led him to publish Three Visits to Madagascar (1858), Madagascar Revisited (1867), and the Martyr Church of Madagascar (1870).
Ellis Island, a small island in the upper New York Bay. Sold by the New York State to the United States, in 1808, it was for a long time used as a powder magazine, but in 1891 was made an immigrant station.
Ello´ra, or Elo´ra, a ruined village, Hindustan, Deccan, Nizam's Dominions, 13 miles north-west of Aurangabad, famous for its rock and cave temples excavated in the crescent-shaped scarp of a large plateau. They run from north to south for about a mile and a quarter, and consist of five Jain caves towards the north, seventeen Brahmanical caves at the centre, and towards the south twelve Buddhistic caves. Of the temples some are cut down through the rock, and left open above like isolated buildings, others are excavated under the hill in the manner of caves properly so called. The interior walls are often richly carved with mythological designs. The most magnificent of the whole is the Hindu temple called Kailasa or Cailasa, the central portion of which forms an isolated excavated mass or immense block 500 feet in circumference and 100 feet high. It is surrounded by galleries or colonnades at the distance of 150 feet, in which
the whole Hindu pantheon is cut in the perpendicular rock. Another fine temple, much smaller, but cut under the hill, is the Buddhist cave of Visvakarma, the only one excavated with a curved roof. The date of the caves is not certainly known, but they were probably the work of the reigning families at the neighbouring Deoghir, now Daulatabad, which, prior to the Mohammedan conquest, A.D. 1293, was the capital of a powerful Hindu principality.—Cf. Fergusson and Burgess, The Cave Temples of India.
Ellore, a town of India, Godavari district, Madras presidency, once the capital of the Northern Circars. Pop. 29,500.