intersects' the transversal; the axis is on a fixed ruled surface to which the plane of the ellipse is a tangent plane, the ellipse being the section of the ruled surface by the plane; the ruled surface is a cylindroid deformed by a simple shear parallel to the transversal. In the third-order complex the centre locus becomes a finite closed quartic surface, with three (one always real) intersecting nodal axes, every plane section of which is a trinodal quartic. The chief defect of the geometrical properties of these bi-quaternions is that the ordinary algebraic scalar finds no place among them, and in consequence Q'1 is meaningless.
Putting I"7]=E we get Combebiac's tri-quaternion under the form Q=£p-I-1;q+wr. This has a reciprocal Q'1=£p"'=17q“ -w]>“1rq", and a conjugate KQ (such that K[QQ']= KQ'KQ, K[KQ]=Q) given by KQ=§ Kq+r;Kp+wK1'; the product QQ' of Q and Q' is Zpp'-l-17qq'+w(p1'-l-rq'); the quasi-vector § (1-K)Q is Combebiac's linear element and may be regarded as a point on a line; the quasi-scalar (in a different sense from the rest of this article) § (1-l-K)Q is Cornbebiac's scalar (Sj>-I-Sq)-I-Combebiads plane. Combebiac does not use K; and in place of 5,17 he uses }, L=7]"E, so that /J.Z=I, (.0}.l.= -pw =w, w2=o. Combebiac's tri-quaternion may be regarded from many simplifying points of view. Thus, in place of his general tri-quaternion we might deal with products of an odd number of point-plane-scalars (of form pq-l-wr) which are themselves point-plane»scalars; and products of an even number which are octonions; the quotient of two point-plane-scalars would be an octonion, of two octonions an octonion, of an octonion by a point-plane-scalar or the inverse a point-plane-scalar. Again a unit point pi may be regarded as by multiplication changing (a) from octonion to point-plane-scalar, (11) from point-plane-scalar to octonion, (c) from plane-scalar to linear element, (d) from linear element to plane-scalar. 1 If -Q= EP-H1fI+wf and W6 Put Q=(I+%w¢)(£P+11<1)>< (1 -l-%wt)'1 we find that the quaternion t must be 2f(1')/f(q-12), where f(r)=rq-Kpr. The point p=Vt may be called the centre of Q and the length St may be called the radius. If Q and Q' are commutative, that is, if QQ'1=Q'Q, then Q and Q' have the same centre and the same radius. Thus Q", Q, Q', Q3, . . . have a common centre and common radius. Q and KQ have a common centre and equal and opposite radii; that is, the t of KQ is the negative conjugate of that of Q. When Su=o, (I'i'%(.|.)1l) () (I-l-%(.0ll) '1 is an operator which shifts (without further change) the tri-quaternion operand an amount given by n in direction and distance.
Bibliography.—In 1904 Alexander Macfarlane published a Bibliography of Quaternions and allied systems of Mathematics for the International Association for promoting the study of Quaternions and allied systems of Mathematics (Dublin University Press); the pamphlet contains 86 pages. In 1899 and 1901 Sir W. R. Hamilton's classical Elements of Quaternions of 1866 was republished under C. J. Joly's editorship, in two volumes (London). Joly adds valuable notes and thirteen important appendices. In 1890 the 3rd edition of P. G. Tait's Elementary Treatise on Quaternions appeared (Cambridge). In 1905 C. J. Joly published his Manual of Quaternions (London); the valuable contents of this are doubled by copious so-called examples; every earnest student should take these as part of the main treatise. The above three treatises may be regarded as the great storehouses; the handling of the subject is very different in the three. The following should also be mentioned: A. McAulay, Octonions, a development of Clifford's Bi-quaternions (Cambridge, 1898); G. Combebiac, Calcul des triquaternions (Paris, 1902); Don Francisco Pérez de Munoz, Introduccion al estudio del calculo de Cuaterniones y otras Algebras especiales (Madrid, 1905); A. McAulay, Algebra after Hamilton, or Multenions (Edinburgh, 1908). (A. McA.)
QUATORZAIN (from Fr. qnatorze, fourteen), the term used in English literature, as opposed to “ sonnet, ” for a poem in fourteen rhymed iambic lines closing (as a sonnet strictly never does) with a couplet. The distinction was long neglected, because the English poets of the 16th century had failed to apprehend the true form of the sonnet, and called Petrarch's and other Italian poets' sonnets quatorzains, and their own incorrect quatorzains sonnets. Almost all the so-called sonnets of the Elizabethan cycles, including those of Shakespeare, Sidney, Spenser and Daniel, are really quatorzains. They consist of three quatrains of alternate rhyme, not repeated in the successive quatrains, and the whole closes with a couplet. A more perfect example of the form could hardly be found than the following, published by Michael Drayton in 1602:- Dear why should you commend me to my rest,
When now the night doth summon all to sleep? Methinks this time becometh lovers best,
Night was ordained together friends to keep. How happy are all other living things
Which though the day conjoin by several flight, The quiet evening yet together brings,
And each returns unto his love at night,
0 thou that art so courteous unto all,
Why should'st thou, Night, abuse me only thus, That every creature to his kind dost call,
And yet 'tis thou dost only sever us?
Well could I wish it would be ever day,
If, when night comes, you bid me go away.
Donne, and afterwards Milton, fought against the facility and incorrectness of this form of metre and adopted the Italian form of sonnet. During the 10th century, most poets of distinction prided themselves on following the strict Petrarchan model of the sonnet, and particularly in avoiding the final couplet. In his most mature period, however, Keats returned to the quatorzain, perhaps in emulation with Shakespeare; and some of his examples, such as “When I have fears, ” “Standing aloof in giant ignorance, ” and “ Bright Star, ” are the most beautiful in modern literature. The “ Fancy in Nubibus, ” written by S. T. Coleridge in 1819, also deserves notice as a quatorzain of peculiar beauty.
QUATRAIN, sometimes spelt Qnartain (from Fr. quatre, four), a piece of verse complete in four rhymed lines. The length or measure .of the verse is immaterial, but they must be bound together by a rhyme-arrangement. This form has always been popular for use in the composition of epigrams, on account of its brevity and neatness, and may be considered as a modification of the Greek or Latintepigram at its concisest.
QUATREFAGES DE BRÉAU, JEAN LOUIS ARMAND DE (1810–1892), French naturalist, was born at Berthezène, near Vallerangue (Gard), on the 10th of February 1810, the son of a Protestant farmer. He studied medicine at Strassburg, where he took the double degree of M.D. and D.Sc., one of his theses being a Théorie d'un coup de canon (November 1829); next year he published a book, Sur les aérolithes, and in 1832 a treatise on L'Extraversion de la vessie. Removing to Toulouse, he practised medicine for a short time, and contributed various memoirs to the local Journal de médecine and to the Annales des sciences naturelles (1834º36). But being unable to continue his researches in the provinces, he resigned the chair of zoology to which he had been appointed, and in 1839 settled in Paris, where he found in H. Milne-Edwards a patron and a friend. Elected professor of natural history at the Lycée Napoléon in 18 50, he 'became a member of the Academy of Sciences in 1852, and in 1855 was called to the chair of anthropology and ethnography at the Musée d'histoire naturelle. Other distinctions followed rapidly, and continued to the end of his otherwise uneventful career, the more important being honorary member of the Royal Society of London (June 1879), member of the Institute and of the Académie de médecine, and commander of the Legion of Honour (1881). He died in Paris on the 12th of January 1892. He was an accurate observer and unwearied collector of zoological materials, gifted with remarkable descriptive power, and possessed of a clear, vigorous style, but somewhat deficient in deep philosophic insight. Hence his serious studies on the anatomical characters of the lower and higher organisms, man included, will retain their value, while many of his theories and generalizations, especially in the department of ethnology, are already forgotten.
The work of de Quatrefages ranged over the whole field of zoology from the annelids and other low organisms to the anthropoids and man. Of his numerous essays in scientific periodicals, the. more important were: Considérations sur les caractères zoologiques des rongeurs (1840); “ De l'organisation des animaux sans vertèbres des Còtes de la Manche ” (Ann. Sc. Nat., 1844); “ Recherches sur