Part 20

FUMAROLE, a vent from which volcanic vapours issue, named indirectly from the Lat. _fumariolum_, a smoke-hole. The vapours from fumaroles were studied first by R.W. Bunsen, on his visit to Iceland, and afterwards by H. Sainte-Claire Deville and other chemists and geologists in France, who examined the vapours from Santorin, Etna, &c. The hottest vapours issue from dry fumaroles, at temperatures of at least 500 deg. C., and consist chiefly of anhydrous chlorides, notably sodium chloride. The acid fumaroles yield vapours of lower temperature (300 deg. to 400 deg.) containing much water vapour, with hydrogen chloride and sulphur dioxide. The alkaline fumaroles are still cooler, though above 100 deg., and evolve ammonium chloride with other vapours. Cold fumaroles, below 100 deg., discharge principally aqueous vapour, with carbon dioxide, and perhaps hydrogen sulphide. The fumaroles of Mont Pele in Martinique during the eruption of 1902 were examined by A. Lacroix, and the vapours analysed by H. Moissan, who found that they consisted chiefly of water vapour, with hydrogen chloride, sulphur, carbon dioxide, carbon monoxide, methane, hydrogen, nitrogen, oxygen and argon. These vapours issued at a temperature of about 400 deg. Armand Gautier has pointed out that these gases are practically of the same composition as those which he obtained on heating granite and certain other rocks. (See VOLCANO).

FUMIGATION (from Lat. _fumigare_, to smoke), the process of producing smoke or fumes, as by burning sulphur, frankincense, tobacco, &c., whether as a ceremony of incantation, or for perfuming a room, or for purposes of disinfection or destruction of vermin. In medicine the term has been used of the exposure of the body, or a portion of it, to fumes such as those of nitre, sal-ammoniac, mercury, &c.; fumigation, by the injection of tobacco smoke into the great bowel, was a recognized procedure in the 18th century for the resuscitation of the apparently drowned. "Fumigated" or "fumed" oak is oak which has been darkened by exposure to ammonia vapour.

FUMITORY, in botany, the popular name for the British species of _Fumaria_, a genus of small, branched, often climbing annual herbs with much-divided leaves and racemes of small flowers. The flowers are tubular with a spurred base, and in the British species are pink to purplish in colour. They are weeds of cultivation growing in fields and waste places. _F. capreolata_ climbs by means of twisting petioles. In past times fumitory was in esteem for its reputed cholagogue and other medicinal properties; and in England, boiled in water, milk or whey, it was used as a cosmetic. The root of the allied species (_Corydalis cava_ or _tuberosa_) is known as _radix aristolochia_, and has been used medicinally for various cutaneous and other disorders, in doses of 10 to 30 grains. Some eleven alkaloids have been isolated from it. The herbage of _Fumaria officinalis_ and _F. racemosa_ is used in China under the name of _Tsze-hwa-ti-ting_ as an application for glandular swellings, carbuncles and abscesses, and was formerly valued in jaundice, and in cases of accidental swallowing of the beard of grain (see F. Porter Smith, _Contrib. towards the Mat. Medica ... of China_, p. 99, 1871). The name fumitory, Latin _fumus terrae_, has been supposed to be derived from the fact that its juice irritates the eyes like smoke (see Fuchs, _De historia stirpium_, p. 338, 1542); but _The Grete Herball_, cap. clxix., 1529, fol., following the _De simplici medicina_ of Platearius, fo. xciii. (see in _Nicolai Praepositi dispensatorium ad aromatarios_, 1536), says: "It is called Fumus terre fume or smoke of the erthe bycause it is engendred of a cours fumosyte rysynge frome the erthe in grete quantyte lyke smoke: this grosse or cours fumosyte of the erthe wyndeth and wryeth out: and by workynge of the ayre and sonne it turneth into this herbe."

FUNCHAL, the capital of the Portuguese archipelago of the Madeiras; on the south coast of Madeira, in 32 deg. 37' N. and 16 deg. 54' W. Pop. (1900) 20,850. Funchal is the see of a bishop, in the archiepiscopal province of Lisbon; it is also the administrative centre of the archipelago, and the residence of the governor and foreign consuls. The city has an attractive appearance from the sea. Its whitewashed houses, in their gardens full of tropical plants, are built along the curving shore of Funchal Bay, and on the lower slopes of an amphitheatre of mountains, which form a background 4000 ft. high. Numerous country houses (_quintas_), with terraced gardens, vineyards and sugar-cane plantations occupy the surrounding heights. Three mountain streams traverse the city through deep channels, which in summer are dry, owing to the diversion of the water for irrigation. A small fort, on an isolated rock off shore, guards the entrance to the bay, and a larger and more powerfully armed fort crowns an eminence inland. The chief buildings include the cathedral, Anglican and Presbyterian churches, hospitals, opera-house, museum and casino. There are small public gardens and a meteorological observatory. In the steep and narrow streets, which are lighted by electricity, wheeled traffic is impossible; sledges drawn by oxen, and other primitive conveyances are used instead (see MADEIRA). In winter the fine climate and scenery attract numerous invalids and other visitors, for whose accommodation there are good hotels; many foreigners engaged in the coal and wine trades also reside here permanently. The majority of these belong to the British community, which was first established here in the 18th century. Funchal is the headquarters of Madeiran industry and commerce (see MADEIRA). It has no docks and no facilities for landing passengers or goods; vessels are obliged to anchor in the roadstead, which, however, is sheltered from every wind except the south. Funchal is connected by cable with Carcavellos (for Lisbon), Porthcurnow (for Falmouth, England) and St Vincent in the Cape Verde Islands (for Pernambuco, Brazil).

FUNCTION,[1] in mathematics, a variable number the value of which depends upon the values of one or more other variable numbers. The theory of functions is conveniently divided into (I.) Functions of Real Variables, wherein real, and only real, numbers are involved, and (II.) Functions of Complex Variables, wherein complex or imaginary numbers are involved.

I. FUNCTIONS OF REAL VARIABLES

1. _Historical._--The word function, defined in the above sense, was introduced by Leibnitz in a short note of date 1694 concerning the construction of what we now call an "envelope" (_Leibnizens mathematische Schriften_, edited by C.I. Gerhardt, Bd. v. p. 306), and was there used to denote a variable length related in a defined way to a variable point of a curve. In 1698 James Bernoulli used the word in a special sense in connexion with some isoperimetric problems (Joh. Bernoulli, _Opera_, t. i. p. 255). He said that when it is a question of selecting from an infinite set of like curves that one which best fulfils some function, then of two curves whose intersection determines the thing sought one is always the "line of the function" (_Linea functionis_). In 1718 John Bernoulli (_Opera_, t. ii. p. 241) defined a "function of a variable magnitude" as a quantity made up in any way of this variable magnitude and constants; and in 1730 (Opera, t. iii. p. 174) he noted a distinction between "algebraic" and "transcendental" functions. By the latter he meant integrals of algebraic functions. The notation [f](x) for a function of a variable x was introduced by Leonhard Euler in 1734 (_Comm. Acad. Petropol._ t. vii. p. 186), in connexion with the theorem of the interchange of the order of differentiations. The notion of functionality or functional relation of two magnitudes was thus of geometrical origin; but a function soon came to be regarded as an analytical expression, not necessarily an algebraic expression, containing the variable or variables. Thus we may have rational integral algebraic functions such as _ax_^2 + _bx_ + c, or rational algebraic functions which are not integral, such as

a1x^n + a2x^(n - 1) + ... + a_n -------------------------------, b1x^m + b2x^(m - 1) + ... + b_m

or irrational algebraic functions, such as [root]x, or, more generally the algebraic functions that are determined implicitly by an algebraic equation, as, for instance,

[f]_n(x, y) + [f]_(n - 1) (x, y) + ... + [f]0 = 0

where [f]_n(x, y), ... mean homogeneous expressions in x and y having constant coefficients, and having the degrees indicated by the suffixes, and [f]0 is a constant. Or again we may have trigonometrical functions, such as sin x and tan x, or inverse trigonometrical functions, such as sin^(-1)x, or exponential functions, such as e^x and a^x, or logarithmic functions, such as log x and log (1 + x). We may have these functional symbols combined in various ways, and thus there arises a great number of functions. Further we may have functions of more than one variable, as, for instance, the expression xy/(x^2 + y^2), in which both x and y are regarded as variable. Such functions were introduced into analysis somewhat unsystematically as the need for them arose, and the later developments of analysis led to the introduction of other classes of functions.

2. _Graphic Representation._--In the case of a function of one variable x, any value of x and the corresponding value y of the function can be the co-ordinates of a point in a plane. To any value of x there corresponds a point N on the axis of x, in accordance with the rule that x is the abscissa of N. The corresponding value of y determines a point P in accordance with the rule that x is the abscissa and y the ordinate of P. The ordinate y gives the value of the function which corresponds to that value of the variable x which is specified by N; and it may be described as "the value of the function at N." Since there is a one-to-one correspondence of the points N and the numbers x, we may also describe the ordinate as "the value of the function at x." In simple cases the aggregate of the points P which are determined by any

## particular function (of one variable) is a curve, called the "graph of

the function" (see S 14). In like manner a function of two variables defines a surface.

3. _The Variable._--Graphic methods of representation, such as those just described, enabled mathematicians to deal with irrational values of functions and variables at the time when there was no theory of irrational numbers other than Euclid's theory of incommensurables. In that theory an irrational number was the ratio of two incommensurable geometric magnitudes. In the modern theory of number irrational numbers are defined in a purely arithmetical manner, independent of the measurement of any quantities or magnitudes, whether geometric or of any other kind. The definition is effected by means of the system of _ordinal_ numbers (see NUMBER). When this formal system is established, the theory of measurement may be founded upon it; and, in particular, the co-ordinates of a point are defined as numbers (not lengths), which are assigned in accordance with a rule. This rule involves the measurement of lengths. The theory of functions can be developed without any reference to graphs, or co-ordinates or lengths. The process by which analysis has been freed from any consideration of measurable quantities has been called the "arithmetization of analysis." In the theory so developed, the variable upon which a function depends is always to be regarded as a number, and the corresponding value of the function is also a number. Any reference to points or co-ordinates is to be regarded as a picturesque mode of expression, pointing to a possible application of the theory to geometry. The development of "arithmetized analysis" in the 19th century is associated with the name of Karl Weierstrass.

All possible values of a variable are numbers. In what follows we shall confine our attention to the case where the numbers are real. When complex numbers are introduced, instead of real ones, the theory of functions receives a wide extension, which is accompanied by appropriate limitations (see below, II. Functions of Complex Variables). The set of all real numbers forms a _continuum_. In fact the notion of a one-dimensional continuum first becomes precise in virtue of the establishment of the system of real numbers.

4. _Domain of a Variable.--Theory of Aggregates._--The notion of a "variable" is that of a number to which we may assign at pleasure any one of the values that belong to some chosen set, or _aggregate_, of numbers; and this set, or aggregate, is called the "domain of the variable." This domain may be an "interval," that is to say it may consist of two terminal numbers, all the numbers between them and no others. When this is the case the number is said to be "continuously variable." When the domain consists of all real numbers, the variable is said to be "unrestricted." A domain which consists of all the real numbers which exceed some fixed number may be described as an "interval unlimited towards the right"; similarly we may have an interval "unlimited towards the left."

In more complicated cases we must have some rule or process for assigning the aggregate of numbers which constitute the domain of a variable. The methods of definition of particular types of aggregates, and the theorems relating to them, form a branch of analysis called the "theory of aggregates" (_Mengenlehre, Theorie des ensembles, Theory of sets of points_). The notion of an "aggregate" in general underlies the system of ordinal numbers. An aggregate is said to be "infinite" when it is possible to effect a one-to-one correspondence of all its elements to some of its elements. For example, we may make all the integers correspond to the even integers, by making 1 correspond to 2, 2 to 4, and generally n to 2n. The aggregate of positive integers is an infinite aggregate. The aggregates of all rational numbers and of all real numbers and of points on a line are other examples of infinite aggregates. An aggregate whose elements are real numbers is said to "extend to infinite values" if, after any number N, however great, is specified, it is possible to find in the aggregate numbers which exceed N in absolute value. Such an aggregate is always infinite. The "neighbourhood of a number (or point) a for a positive number h" is the aggregate of all numbers (or points) x for which the absolute value of x - a denoted by |x - a|, does not exceed h.

5. _General Notion of Functionality._--A function of one variable was for a long time commonly regarded as the ordinate of a curve; and the two notions (1) that which is determined by a curve supposed drawn, and (2) that which is determined by an analytical expression supposed written down, were not for a long time clearly distinguished. It was for this reason that Fourier's discovery that a single analytical expression is capable of representing (in different parts of an interval) what would in his time have been called different functions so profoundly struck mathematicians (S 23). The analysts who, in the middle of the 19th century, occupied themselves with the theory of the convergence of Fourier's series were led to impose a restriction on the character of a function in order that it should admit of such representation, and thus the door was opened for the introduction of the general notion of functional dependence. This notion may be expressed as follows: We have a variable number, y, and another variable number, x, a domain of the variable x, and a rule for assigning one or more definite values to y when x is any point in the domain; then y is said to be a "function" of the variable x, and x is called the "argument" of the function. According to this notion a function is, as it were, an indefinitely extended table, like a table of logarithms; to each point in the domain of the argument there correspond values for the function, but it remains arbitrary what values the function is to have at any such point.

For the specification of any particular function two things are requisite: (1) a statement of the values of the variable, or of the aggregate of points, to which values of the function are to be made to correspond, i.e. of the "domain of the argument"; (2) a rule for assigning the value or values of the function that correspond to any point in this domain. We may refer to the second of these two essentials as "the rule of calculation." The relation of functions to analytical expressions may then be stated in the form that the rule of calculation is: "Give the function the value of the expression at any point at which the expression has a determinate value," or again more generally, "Give the function the value of the expression at all points of a definite aggregate included in the domain of the argument." The former of these is the rule of those among the earlier analysts who regarded an analytical expression and a function as the same thing, and their usage may be retained without causing confusion and with the advantage of brevity, the analytical expression serving to specify the domain of the argument as well as the rule of calculation, e.g. we may speak of "the function 1/x." This function is defined by the analytical expression 1/x at all points except the point x = 0. But in complicated cases separate statements of the domain of the argument and the rule of calculation cannot be dispensed with. In general, when the rule of calculation is determined as above by an analytical expression at any aggregate of points, the function is said to be "represented" by the expression at those points.

When the rule of calculation assigns a single definite value for a function at each point in the domain of the argument the function is "uniform" or "one-valued." In what follows it is to be understood that all the functions considered are one-valued, and the values assigned by the rule of calculation real. In the most important cases the domain of the argument of a function of one variable is an interval, with the possible exception of isolated points.

6. _Limits._--Let [f](x) be a function of a variable number x; and let a be a point such that there are points of the domain of the argument x in the neighbourhood of a for any number h, however small. If there is a number L which has the property that, after any positive number [epsilon], however small, has been specified, it is possible to find a positive number h, so that |L - [f](x)| < [epsilon] for all points x of the domain (other than a) for which |x-a| < h, then L is the "limit of [f](x) at the point a." The condition for the existence of L is that, after the positive number [epsilon] has been specified, it must be possible to find a positive number h, so that |[f](x') - [f](x)| < [epsilon] for all points x and x' of the domain (other than a) for which |x - a| < h and |x' - a| < h.

It is a fundamental theorem that, when this condition is satisfied, there exists a perfectly definite number L which is the limit of [f](x) at the point a as defined above. The limit of [f](x) at the point a is denoted by Lt_(x = a)[f](x), or by lim_(x = a)[f](x).

If [f](x) is a function of one variable x in a domain which extends to infinite values, and if, after [epsilon] has been specified, it is possible to find a number N, so that |[f](x') - [f](x)| <[epsilon] for all values of x and x' which are in the domain and exceed N, then there is a number L which has the property that |[f](x) - L| < [epsilon] for all such values of x. In this case [f](x) has a limit L at x = [oo]. In like manner [f](x) may have a limit at x = -[oo]. This statement includes the case where the domain of the argument consists exclusively of positive integers. The values of the function then form a "sequence," u1, u2, ... u_n, ..., and this sequence can have a limit at n = [oo].

The principle common to the above definitions and theorems is called, after P. du Bois Reymond, "the general principle of convergence to a limit."

It must be understood that the phrase "x = [oo]" does not mean that x takes some particular value which is infinite. There is no such value. The phrase always refers to a limiting process in which, as the process is carried out, the variable number x increases without limit: it may, as in the above example of a sequence, increase by taking successively the values of all the integral numbers; in other cases it may increase by taking the values that belong to any domain which "extends to infinite values."

A very important type of limits is furnished by _infinite series_. When a sequence of numbers u1, u2, ... u_n, ... is given, we may form a new sequence s1, s2, ... s_n, ... from it by the rules s1 = u1, s2 = u1 + u2, ... s_n = u1 + u2 + ... + u_n or by the equivalent rules s1 = u, s_n - s_(n - 1) = u_n(n = 2, 3, ...). If the new sequence has a limit at n = [oo], this limit is called the "sum of the infinite series" u1 + u2 + ..., and the series is said to be "convergent" (see SERIES).

A function which has not a limit at a point a may be such that, if a certain aggregate of points is chosen out of the domain of the argument, and the points x in the neighbourhood of a are restricted to belong to this aggregate, then the function has a limit at a. For example, sin(1/x) has limit zero at 0 if x is restricted to the aggregate 1/[pi], 1/2[pi], ... 1/n[pi], ... or to the aggregate 1/2[pi], 2/5[pi], ... n/(n^2 + 1)[pi], ..., but if x takes all values in the neighbourhood of 0, sin (1/x) has not a limit at 0. Again, there may be a limit at a if the points x in the neighbourhood of a are restricted by the condition that x - a is positive; then we have a "limit on the right" at a; similarly we may have a "limit on the left" at a point. Any such limit is described as a "limit for a restricted domain." The limits on the left and on the right are denoted by [f](a - 0) and [f](a + 0).

The limit L of [f](x) at a stands in no necessary relation to the value of [f](x) at a. If the point a is in the domain of the argument, the value of [f](x) at a is assigned by the rule of calculation, and may be different from L. In case [f](a) = L the limit is said to be "attained." If the point a is not in the domain of the argument, there is no value for [f](x) at a. In the case where [f](x) is defined for all points in an interval containing a, except the point a, and has a limit L at a, we may arbitrarily annex the point a to the domain of the argument and assign to [f](a) the value L; the function may then be said to be "extrinsically defined." The so-called "indeterminate forms" (see INFINITESIMAL CALCULUS) are examples.