of which the co-ordinates are meromorphic functions of two variables
of the simplest kind, with four sets of periods, is characterized by
; or again, any surface possessing a linear system of
curves of which the order exceeds twice the deficiency of the individual
curves diminished by two, is reducible by birational transformation
to a ruled surface or is a rational surface. But beyond
the general statement that much progress has already been made
in this direction, of great interest to the student of the theory of
functions, nothing further can be added here.
Bibliography.—The learner will find a lucid introduction to the theory in E. Goursat, Cours d’analyse mathématique, t. ii. (Paris, 1905), or, with much greater detail, in A. R. Forsyth, Theory of Functions of a Complex Variable (2nd ed., Cambridge, 1900); for logical rigour in the more difficult theorems, he should consult W. F. Osgood, Lehrbuch der Functionentheorie, Bd. i. (Leipzig, 1906–1907); for greater precision in regard to the necessary quasi-geometrical axioms, beside the indications attempted here, he should consult W. H. Young, The Theory of Sets of Points (Cambridge, 1906), chs. viii.-xiii., and C. Jordan, Cours d’analyse, t. i. (Paris, 1893), chs. i., ii.; a comprehensive account of the Theory of Functions of Real Variables is by E. W. Hobson (Cambridge, 1907). Of the theory regarded as based after Weierstrass upon the theory of power series, there is J. Harkness and F. Morley, Introduction to the Theory of Analytic Functions (London, 1898), an elementary treatise; for the theory of the convergence of series there is also T. J. I’A. Bromwich, An Introduction to the Theory of Infinite Series (London, 1908); but the student should consult the collected works of Weierstrass (Berlin, 1894 ff.), and the writings of Mittag-Leffler in the early volumes of the Acta mathematica; earlier expositions of the theory of functions on the basis of power series are in C. Méray, Leçons nouvelles sur l’analyse infinitésimale (Paris, 1894), and in Lagrange’s books on the Theory of Functions. An account of the theory of potential in its applications to the present theory is found in most treatises; in particular consult E. Picard, Traité d’analyse, t. ii. (Paris, 1893). For elliptic functions there is an introductory book, P. Appell and E. Lacour, Principes de la théorie des fonctions elliptiques et applications (Paris, 1897), beside the treatises of G. H. Halphen, Traité des fonctions elliptiques et de leurs applications (three parts, Paris, 1886 ff.), and J. Tannery et J. Molk, Éléments de la théorie des fonctions elliptiques (Paris, 1893 ff.); a book, A. G. Greenhill, The Applications of Elliptic Functions (London, 1892), shows how the functions enter in problems of many kinds. For modular functions there is an extensive treatise, F. Klein and R. Fricke, Theorie der elliptischen Modulfunctionen (Leipzig, 1890); see also the most interesting smaller volume, F. Klein, Über das Ikosaeder (Leipzig, 1884) (also obtainable in English). For the theory of Riemann’s surface, and algebraic integrals, an interesting introduction is P. Appeil and E. Goursat, Théorie des fonctions algébriques et de leurs intégrales; for Abelian functions see also H. Stahl, Theorie der Abel’schen Functionen (Leipzig, 1896), and H. F. Baker, An Introduction to the Theory of Multiply Periodic Functions (Cambridge, 1907), and H. F. Baker, Abel’s Theorem and the Allied Theory, including the Theory of the Theta Functions (Cambridge, 1897); for theta functions of one variable a standard work is C. G. Jacobi, Fundamenta nova, &c. (Königsberg, 1828); for the general theory of theta functions, consult W. Wirtinger, Untersuchungen über Theta-Functionen (Leipzig, 1895). For a history of the theory of algebraic functions consult A. Brill and M. Noether, Die Entwicklung der Theorie der algebraischen Functionen in älterer und neuerer Zeit, Bericht der deutschen Mathematiker-Vereinigung (1894); and for a special theory of algebraic functions, K. Hensel and G. Landsberg, Theorie der algebraischen Function u.s.w. (Leipzig, 1902). The student will, of course, consult also Riemann’s and Weierstrass’s Ges. Werke. For the applications to geometry in general an important contribution, of permanent value, is E. Picard and G. Simart, Théorie des fonctions algébriques de deux variables indépendantes (Paris, 1897–1906). This work contains, as Note v. t. ii. p. 485, a valuable summary by MM. Castelnuovo and Enriques, Sur quelques résultats nouveaux dans la théorie des surfaces algébriques, containing many references to the numerous memoirs to be found, for the most part, in the transactions of scientific societies and the mathematical journals of Italy.
Beside the books above enumerated there exists an unlimited number of individual memoirs, often of permanent importance and only imperfectly, or too elaborately, reproduced in the pages of the volumes in which the student will find references to them. The German Encyclopaedia of Mathematics, and the Royal Society’s Reference Catalogue of Current Scientific Literature, Pure Mathematics, published yearly, should also be consulted. (H. F. Ba.)
FUNDY, BAY OF, an inlet of the North Atlantic, separating
New Brunswick from Nova Scotia. It is 145 m. long and 48 m.
wide at the mouth, but gradually narrows towards the head,
where it divides into Chignecto Bay to the north, which subdivides
into Shepody Bay and Cumberland Basin (the French
Beaubassin), and Minas Channel, leading into Minas Basin, to
the east and south. Off its western shore opens Passamaquoddy
Bay, a magnificent sheet of deep water with good anchorage,
receiving the waters of the St Croix river and forming part of
the boundary between New Brunswick and the state of Maine.
The Bay of Fundy is remarkable for the great rise and fall of
the tide, which at the head of the bay has been known to reach
62 ft. In Passamaquoddy Bay the rise and fall is about 25 ft.,
which gradually increases toward the narrow upper reaches.
At spring tides the water in the Bay of Fundy is 19 ft. higher
than it is in Bay Verte, in Northumberland Strait, only 15 m.
distant. Though the bay is deep, navigation is rendered
dangerous by the violence and rapidity of the tide, and in summer
by frequent fogs. At low tide, at such points as Moncton or
Amherst, only an expanse of red mud can be seen, and the tide
rushes in a bore or crest from 3 to 6 ft. in height. Large areas
of fertile marshes are situated at the head of the bay, and the
remains of a submerged forest show that the land has subsided
in the latest geological period at least 40 ft. The bay receives
the waters of the St Croix and St John rivers, and has numerous
harbours, of which the chief are St Andrews (on Passamaquoddy
Bay) and St John in New Brunswick, and Digby and Annapolis
(on an inlet known as Annapolis Basin) in Nova Scotia. It was
first explored by the Sieur de Monts (d. c. 1628) in 1604 and
named by him La Baye Française.
FUNERAL RITES, the ceremonies associated with different
methods of disposing of the dead. (See also Burial and Burial Acts;
Cemetery; and Cremation.) In general we have little
record, except in their tombs, of races which, in a past measured
not merely by hundreds but by thousands of years, occupied
the earth; and exploration of these often furnishes our only
clue to the religions, opinions, customs, institutions and arts of
long vanished societies. In the case of the great culture folks
of antiquity, the Babylonians, Egyptians, Hindus, Persians,
Greeks and Romans, we have, besides their monuments, the
evidence of their literatures, and so can know nearly as much of
their rites as we do of our own. The rites of modern savages
not only help us to interpret prehistoric monuments, but explain
peculiarities in our own rituals and in those of the culture folks
of the past of which the significance was lost or buried under
etiological myths. We must not then confine ourselves to the
rites of a few leading races, neglecting their less fortunate
brethren who have never achieved civilization. It is better to
try to classify the rites of all races alike according as they embody
certain leading conceptions of death, certain fears, hopes, beliefs
entertained about the dead, about their future, and their relations
with the living.
The main ideas, then, underlying funeral rites may roughly be enumerated as follows:
1. The pollution or taboo attaching to a corpse.
2. Mourning.
3. The continued life of the dead as evinced in the housing and equipment of the dead, in the furnishing of food for them, and in the orientation and posture assigned to the body.
4. Communion with the dead in a funeral feast and otherwise.
5. Sacrifice for the dead and expiation of their sins.
6. Death witchery.
7. Protection of the dead from ghouls.
8. Fear of ghosts.
1. A dead body is unclean, and the uncleanness extends to things and persons which touch it. Hence the Jewish law (Num. v. 2) enacted that “whoever is unclean by the dead shall be put outside the camp, that they defile not the camp in the midst whereof the Lord dwells.” Such persons were unclean until the even, and might not eat of the holy things unless they bathed their flesh in water. A high priest might on no account “go in to any dead body” (Lev. xxi. 11). Why a corpse is so widely tabooed is not certain; but it is natural to see one reason in the corruption which in warm climates soon sets in. The common experience that where one has died another is likely to do so may also have contributed, though, of course, there was no scientific idea of infection. The old Persian scriptures are full of this taboo. He who has touched a corpse is “powerless in mind, tongue and hand” (Zend Avesta in Sacred Books of the East, pt. i. p. 120), and the paralysis is inflicted by the innumerable drugs or evil spirits which invest a corpse. Fire and earth, being alike creations of the good and pure god
