x^2/a^2 + y^2/b^2 + z^2/c^2 = 1.
Here a, b, c, are the lengths of the principal semi-axes. A plane section
is an ellipse, but there are two particular directions of the section for
which the ellipse reduces to a circle.
The ellipsoid has important applications in dynamics, as a means of
interpreting algebraic formulae for physical quantities; examples are the
ellipsoids of strain and of gyration, and Poinsot's momental ellipsoid.
Two ellipsoids in which the principal sections lie in the same planes, and
have the same foci, are called _confocal_.
_Ellipsoidal Harmonics_ are mathematical functions by means of which
certain physical problems in heat and electricity relating to ellipsoidal
surfaces can be solved. For ellipsoids of revolution, see _Spheroid_.
ELLIPTIC FUNCTIONS are generalizations of the circular functions _sine_,
_cosine_, &c. If
u = integral from 0 to x of the function {dx}/sqrt(1-x^2) = sin^{-1}x,
we have x = sin u. Similarly, if
u = integral from 0 to x of the function {dx}/sqrt{(1-x^2)(1-k^2x^2)},
we may write x = sn u, [sqrt](1 - x^2) = cn u, [sqrt](1 - k^2x^2) = 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 2[pi]; the
elliptic functions are _doubly_ periodic, having both a real and a pure
imaginary period when k^2 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 - k^2sn^2u sn^2v}.
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.
ELLWOOD, Thomas, an early writer among the Quakers, born in 1639, died in
1713. About 1660 he was induced to join the Society of Friends, and soon
after published _An Alarm to the Priests_. He was imprisoned on account of
his religion, but subsequently became reader to Milton, and upon reading
the MS. of _Paradise Lost_ is said to have suggested to the poet the idea
of writing the _Paradise Regained_. In 1705 and 1709 he published the two
parts of his _Sacred History_. His works include a poetical life of King
David, the _Davideis_, _Forgery no Christianity_, and an autobiography,
_The History of the Life of Thomas Ellwood: written by his own hand._
[Illustration: Common Elm (_Ulmus campestris_)