have

مه

I

==

∞==ƒ_ƒ(x)dx, a2=1ƒ_ƒ(x) cos"** dx, b=f(x) sinxdx.

The interval between -c and c may be called the " 'periodic interval," and we may replace it by any other interval, e.g. that between 0 and 1, without any restriction of generality. When

no. The convergence is said to be "uniform" in an interval | the interval between-c and c, was given by Fourier, viz. we if, after specification of e, the same number n suffices at all points of the interval to make |f(x)-fm(x)|< € for all values of m which exceed n. The numbers n corresponding to any e, however small, are all finite, but, when e is less than some fixed finite number, they may have an infinite superior limit (§ 7); when this is the case there must be at least one point, a, of the interval which has the property that, whatever number N we take, e can be taken so small that, at some point in the neighbourhood of a, n must be taken > N to make [f(x) −fm(x)|<e when m>n; then the series does not converge uniformly in the and this is neighbourhood of a. The distinction may be otherwise expressed thus: Choose a first and e afterwards, then the number n is finite; choose e first and allow a to vary, then the number n becomes a function of a, which may tend to become infinite, or may remain below a fixed number; if such a fixed number exists, however small e may be, the convergence is uniform.

this is done the sum of the series takes the form
f(2) cos(2rx (2-x)) dz,

Li S

sin {(2n+1)(2−x)r}
sin {(2-x)}

dz.

(ii.)

Fourier's theorem is that, if the periodic interval can be divided into a finite number of partial intervals within each of which the function is ordinary (§ 14), the series represents the function within each of those partial intervals. In Fourier's time a For example, the series sin x-sin 2x+} sła 3x-...is conver- function of this character was regarded as completely arbitrary. gent for all real values of x, and, when >>- its sum is *; By a discussion of the integral (ii.) based on the Second Theorem but, when x is but a little less than #, the number of terms which of the Mean (§ 15) it can be shown that, if f(x) has restricted oscilla must be taken in order to bring the sum at all near to the value of tion in the interval (§ 11), the sum of the series is equal to {{f(x+0)+ x is very large, and this number tends to increase indefinitely as f(x-0)) at any point x within the interval, and that it is equal to x approaches 7. This series does not converge uniformly in theƒ(+0)+f(1-0)} at each end of the interval. (See the article neighbourhood of x=x. Another example is afforded by the series FOURIER'S SERIES.) It therefore represents the function at any 2 nx (n+1)x of which the remainder after n terms point of the periodic interval at which the function is continuous n-on2x2+1 (n+1)2x2+1' (except possibly the end-points), and has a definite value at each is nx/(n2x2+1). If we put x=1/n, for any value of n, however point of discontinuity. The condition of restricted oscillation great, the remainder is; and the number of terms required to be includes all the functions contemplated in the statement of the taken to make the remainder tend to zero depends upon the value of theorem and some others. Further, it can be shown that, in any x when x is near to zero-it must, in fact, be large compared with partial interval throughout which f(x) is continuous, the series 1/x. The series does not converge uniformly in the neighbourhood converges uniformly, and that no series of the form (i), with coefficients other than those determined by Fourier's rule, can represent the function at all points, except points of discontinuity, in the same periodic interval. The result can be extended to a function f(x) which tends to become infinite at a finite number of points a of the at each of the points a, (2) the improper definite integral of f(x) interval, provided (1) f(x) tends to become determinately infinite through the interval is convergent, (3) f(x) has not an infinite number of discontinuities or of maxima or minima in the interval.

of x=0.

As regards series whose terms represent continuous functions we have the following theorems:

(1) If the series converges uniformly in an interval it represents a function which is continuous throughout the interval.

(2) If the series represents a function which is discontinuous in an interval it cannot converge uniformly in the interval. (3) A series which does not converge uniformly in an interval may nevertheless represent a function which is continuous throughout the interval.

(4) A power series converges uniformly in any interval contained within its domain of convergence, the end-points being excluded.

(5) If_2_ƒ,(x)=f(x) converges uniformly in the interval

0

between a and b

S.f(x)dx=f.fr(x)dx,

or a series which converges unformly may be integrated term by

term.

24. Representation of Continuous Functions by Series.-If the series for f(x) formed by Fourier's rule converges at the point a of the periodic interval, and if f(x) is continuous at a, the sum of the series is f(a); but it has been proved by P. du Bois Reymond that the function may be continuous at a, and yet the series formed by Fourier's rule may be divergent at a. Thus some continuous functions do not admit of representation by Fourier's series. All continuous functions, however, admit of being represented with arbitrarily close approximation in either of two forms, which may be described as terminated Fourier's series" and "terminated power series," according to the two following theorems:

(1) If f(x) is continuous throughout the intervai between o and 2, and if any positive number e however small is specified, it is possible to find an integer n, so that the difference between the value of f(x) and the sum of the first n terms of the series for f(x), formed by Fourier's rule with periodic interval from o to 27, shall be less than e at all points of the interval. This result can be extended to a function which is continuous in any given interval.

(2) If f(x) is continuous throughout an interval, and any positive number however small is specified, it is possible to find an integer n and a polynomial in x of the nth degree, so that the difference between the value of f(x) and the value of the polynomial shall be less than e at all points of the interval.

Again it can be proved that, if f(x) is continuous throughout a given interval, polynomials in x of finite degrees can be found, so as to form an infinite series of polynomials whose sum is equal to f(x) at all points of the interval. Methods of representation of continuous functions by infinite series of rational fractional

23. Fourier's Series.-An extensive class of functions admit functions have also been devised. of being represented by series of the form aot

The multiplier of e-te under the sign of integration can be expanded (Paris, 1886); W. H. and G. C. Young, The Theory of Sets of Points in the power series

When the series is integrated term by term, the right-hand member des séries trigonométriques," Crelle, Bd. iv.; P. Du Bois Reymond, of the equation for ~(x) takes the form

This series is divergent; but, if it is stopped at any term, the difference between the sum of the series so terminated and the value of (x) is less than the last of the retained terms. Stirling's formula is obtained by retaining the first term only. Other well-known examples of asymp-d. Taylorschen Lehrsatzes," Math. Ann. Bd. xliv.; Riemann, totic expansions are afforded by the descending series for Bessel's functions. Methods of obtaining such expansions for the solutions of linear differential equations of the second order were investigated by G. G. Stokes (Math, and Phys. Papers, vol. ii. p. 329), and a general theory of asymptotic expansions has been developed by H. Poincaré. A still more general theory of divergent series, and of the conditions in which they can be used, as above, for the purposes of approximate calculation has been worked out by E. Borel. The great merit of asymptotic expansions is that they admit of addition, subtraction, multiplication and division, term by term, in the same way as absolutely convergent series, and they admit also of integration term by term; that is to say, the results of such operations are

II.-FUNCTIONS OF COMPLEX VARIABLES

periods; (§ 23), Geometrical applications of Elliptic Functions, shows that any plane curve of deficiency unity can be expressed by elliptic functions, and gives a geometrical proof of the addition theorem for the function (u); (§ 24), Integrals of Algebraic Functions in connexion with the theory of plane curves, discusses the generalization to curves of any deficiency, (§ 25), Monogenic Functions of several independent variables, describes briefly the beginnings of this theory, with a mention of some fundamental theorems: (§ 26), Multiply-Periodic Functions and the Theory of Surfaces, attempts to show the nature of some problems now being actively pursued.

Beside the brevity necessarily attaching to the account here given of advanced parts of the subject, some of the more clementary results are stated only, without proof, as, for instance: the monogeneity of an algebraic function, no reference being made, moreover, to the cases of differential equations whose integrals are monogenic, that a function possessing an algebraic addition theorem is necessarily an elliptic function (or a particular case of such); that any area can be conformally represented on a half plane, a theorem requiring further much more detailed consideration of the meaning of arca than we have given; while the character and properties, including the connectivity, of a Riemann surface have not been referred to. The theta functions are referred to only once, and the principles of the theory of Abelian Functions have been illustrated only by the developments given for elliptic functions.

§ 1. Complex Numbers.-Complex numbers are numbers of the form x+iy, where x, y are ordinary real numbers, and i is a symbol imagined capable of combination with itself and the ordinary real numbers, by way of addition, subtraction, multiplication and division, according to the ordinary commutative, associative and distributive laws; the symbol i is further such that =-1.

In the preceding section the doctrine of functionality is discussed with respect to real quantities; in this section the theory when complex or imaginary quantities are involved receives treatment. The following abstract explains the arrangement of the subject matter: (§ 1), Complex numbers, states what a complex number is; (§ 2), Plotting of simple expressions involving complex numbers, illustrates the meaning in some simple cases, introducing the notion of conformal representation and proving that an algebraic equation has complex, if not real, roots; (§ 3), Limiling operations, defines certain simple functions of a complex variable which are obtained by passing to a limit, in particular the exponential function, and the generalized logarithm, here denoted by λ(2); (§ 4), Functions of a complex variable in geneṛal, after explaining briefly what is to be understood by a region of the complex plane and by a path, and expounding a logical principle of some importance, gives the accepted definition of a function of a complex variable, establishes the existence of a complex integral, and proves Cauchy's theorem relating thereto; (§ 5), Applications, considers the differentiation and integration of series of functions of a complex variable, proves Laurent's theorem, and establishes the expansion of a function of a complex variable as a power series, leading, in (§ 6), Singular points, to a definition of the region of existence and singular points of a function of a complex variable, and thence, in (§ 7), Monogenic Functions, to what the writer believes to be the simplest definition of a function of a complex variable, that of Weierstrass; (§ 8), Some elementary properties of single valued functions, first discusses the meaning of a pole, proves that a single valued function with only poles is rational, gives Mittag-Leffler's theorem, and Weierstrass's theorem for the primary factors of an integral function. stating generalized forms for these, leading to the theorem of (§ 9), The construction of a monogenic function with a given region of existence, with which is connected (§ 10), Expression of a monogenic function by rational functions in a given region, of which the method is applied in (§ 11), Expression of (1-2)-1by polynomials, to a definite example, used here to obtain (§ 12), An expansion of an arbitrary function by means of a series of polynomials, over a star region, also obtained in the original manner of Mittag-variable 2=x+iy, the distance OP be called 7, and the positive Leffler; (§ 13), Application of Cauchy's theorem to the determination of definite integrals, gives two examples of this method; (§ 14), Doubly Periodic Functions, is introduced at this stage as furnishing an excellent example of the preceding principles. The reader who wishes to approach the matter from the point of view of Integral Calculus should first consult the section (§ 20) below, dealing with Elliptic Integrals; (§ 15), Potential Functions, Conformal representation in general, gives a sketch of the connexion of the theory of potential functions with the theory of conformal representation, enunciating the Schwarz-Christoffel theorem for the representation of a polygon, with the application to the case of an equilateral triangle; (§ 16), Multiple-valued Functions, Algebraic Functions, deals for the most part with algebraic functions, proving the residue theorem, and establishing that an algebraic function has a definite Order; (§ 17), Integrals of Algebraic Functions, enunciating Abel's theorem; (§ 18), Indeterminateness of Algebraic Integrals, deals with the periods 2. Plotting and Properties of Simple Expressions involving associated with an algebraic integral, establishing that for an a Complex Number.-If we put =(-i)/(s+i), and, putting elliptic integral the number of these is two; (§ 19), Reversion of 5 =+in, take a new plane upon which, are rectanguan algebraic integral, mentions a problem considered below in lar co-ordinates, the equations §=(x2+y2-1)/[x2+(y+1)2), detail for an elliptic integral; (§ 20), Elliptic Integrals, considers = -2xy/(x2+(y+1)2] will determine, corresponding to any the algebraic reduction of any elliptic integral to one of three point of the first plane, a point of the second plane. There is standard forms, and proves that the function obtained by the one exception of zi, that is, x=o, y=-1, of which the reversion is single-valued; (§ 21), Modular Functions, gives a corresponding point is at infinity. It can now be easily proved statement of some of the more elementary properties of some that as z describes the real axis in its plane the point describes functions of great importance, with a definition of Automorphic once a circle of radius unity, with centre at =0, and that there Functions, and a hint of the connexion with the theory of linear is a definite correspondence of point to point between points differential equations; (§ 22), A property of integral functions, in the s-plane which are above the real axis and points of the deduced from the theory of modalar functions, proves that there-plane which are interior to this circle; in particular =i cannot be more than one value not assumed by an integral corresponds to = 0. function, and gives the basis of the well-known expression of the modulus of the elliptic functions in terms of the ratio of the

Taking in a plane two rectangular axes Ox, Oy, we assume that every point of the plane is definitely associated with two real numbers x, y (its co-ordinates) and conversely; thus any point of the plane is associated with a single complex number; in particular, for every point of the axis Ox, for which y=0, the associated number is an ordinary real number; the complex numbers thus include the real numbers. The axis Ox is often called the real axis, and the axis Oy the imaginary axis. If P be the point associated with the complex angle less than 27 between Ox and OP be called 0, we may write 2=7(cos +i sin 0); then is called the modulus or absolute value of 2 and often denoted by z and is called the phase or amplitude of 2, and often denoted by ph (3); strictly the phase is ambiguous by additive multiples of 2. If 'x'+iy' be represented by P'. the complex argument +z is represented by a point P' obtained by drawing from P' a line equal to and parallel to OP; the geoof the plane; as, for instance, the equation 2+3+2 involves metrical representation involves for its validity certain properties the possibility of constructing a parallelogram (with OP as diagonal). It is important constantly to bear in mind, what is capable of easy algebraic proof (and geometrically is Euclid's proposition III. 7), that the modulus of a sum or difference of two complex numbers is moduli, and is greater than (or equal to) the difference of their generally less than (and is never greater than) the sum of their moduli; the former statement thus holds for the sum of any number of complex numbers. We shall write E(0) for cos i sin ; it is at once verified that E(ia). E(i8) =E[i(a+B)], so that the phase of a product of complex quantities is obtained by addition of their respective phases.

Moreover, being a rational function of 2, both and are con tinuous differentiable functions of x and y, save when is infinite;

writing = f(x, y) =ƒ(2~iy, y), the fact that this is really independent | be in absolute value less than a real positive quantity M, it can be of y leads at once to aflax+idflay=o, and hence to

at an a
Jx ay' ay

so that is not any arbitrary function of x, y, and when έ is known is determinate save for an additive constant. Also, in virtue of these equations, if, be the values of corresponding to two near values of z, say z and 2', the ratio (-5)/(-2) has a definite limit when '=2, independent of the ultimate phase of 2-s, this limit being therefore equal to 05/0x, that is, d/dx+ion/dx. Geometrically this fact is interpreted by saying that if two curves in the 2-plane intersect at a point P, at which both the differential coefficients o/dx, dŋ/dx are not zero, and P', P' be two points near to P on these curves respectively, and the corresponding points of the -plane be Q, Q, Q, then (1) the ratios PP/PP', QQ′′/QQ' are ultimately equal, (2) the angle P'PP' is equal to Q'Q (3) the rotation from PP' to PP is in the same sense as from QQ to QQ', it being understood that the axes of E, in the one plane are related as are the axes of x, y. Thus any diagram of the z-plane becomes a diagram of the 5-plane with the same angles; the magnification, however, which is equal to point to point. Conversely, it appears subsequently that the expression of any copy of a diagram (say, a map) which preserves angles requires As another illustration consider the case when is a polynomial 5=Poz" + Pizn-1+...+pai

2

(3) varies from

the intervention of the complex variable.

in z,

H being an arbitrary real positive number, it can be shown that a radius R can be found such for every > R we have > H; consider the lower limit of 1 for 12] <R; as +7 is a real continuous function of x, y for <R, there is a point (x, y), say (xo. J), at which is least, say equal to p, and therefore within a circle in the 5-plane whose centre is the origin, of radius p, there are no points representing values corresponding to || <R. But if to be the value of corresponding to (xo, yo), and the expression of near 2x+1yo, in terms of 2-zo. be A(z −20)TM+ B(-20)+1+..., where A is not zero, to two points near to (x, y),

say (xı, yı) or 21 and 21=20+(2−20) (cos +i sin), will corre

m

spond two points near to to, say 1, and 25-, situated so that fo is between them. One of these must be within the circle (p). We infer then that p=0, and have proved that every polynomial in zvanishes for some value of 2, and can therefore be written as a product of factors of the form z-a, where a denotes a complex number. This proposition alone suffices to suggest the importance of complex numbers.

shown that for [2]=r, every term is also less than M in absolute value,
namely, a Mr. If in every arbitrarily small neighbourhood of
zo there be a point for which two converging power series Zan",
if Ea, vanish at z=o there is a circle of finite radius about z=o as
Eb, agree in value, then the series are identical, or a, b,; thus also
centre within which no other points are found for which the sum of
the series is zero. Considering a power series f(z) = Zanz" of radius of
the resulting series Za(2+) may be regarded as a double series
convergence R, if <R and we put + with <R-1201.
in and f, which, since +<R, absolutely convergent;
it may then be arranged according to powers of t. Thus we may
write f(z) =EA!"; hence Ao=f(20), and we have [f(20+ 1) − f(20)]/8 =
ZA, wherein the continuous series on the right reduces to A
for to; thus the ratio on the left has a definite limit when t=0.
equal namely to A, or Enanzo In other words, the original series
may legitimately be differentiated at any interior point zo of its circle
of convergence. Repeating this process we find f(z+1) = Σ!"f{"} (20)/11!,
where f() is the nth differential coefficient. Repeating for this
power series, in, the argument applied about z=0 for 2a,, we
infer that for the series f(z) every point which reduces it to zero is
an isolated point, and of such points only a finite number lie within
a circle which is within the circle of convergence of f(z).
23/3+ of which the radius of convergence is infinite. By
Perhaps the simplest possible power series'ise = exp (2) = 1+22/2!+
multiplication we have exp ().exp (1) = exp (+2'). In particular
when x, y are real, and z=x+iy, exp (z) = exp (x) exp (iy). Now the
functions
Uosin y, Vo-1-cos y, U1y-sin y,

V=4-1+cos y, U2-ly3-y+sin y, V,y-4y2+1-cos y,...
the first is the preceding one; as a function (of a real variable) is
all vanish for yo, and the differential coefficient of any one after
increasing when its differential coefficient is positive, we infer, for
limit we hence infer that
y positive, that each of these functions is positive; proceeding to a

cos y=1-y2+-... sin y=y~}y3+x{ʊys—.........,
for positive, and hence, for all values of y. We thus have exp (iy) =
(z) = exp (x). (cos y+i sin y). In other words,
the modulus of exp

cos y +i sin y, and is exp (x) and the phase is y. Hence also

exp (z+2xi) = exp (x)[cos (y+2x)+i sin (y+2′′)}, which we express by saying that' exp (2) has the period 2ri, and hence also the period akri, where k is an arbitrary integer. From the fact that the constantly increasing function exp (x) can vanish only for x=o, we at once prove that exp (2) has no other periods.

Taking in the plane of s an infinite strip lying between the lines §3. Limiting Operations.-In order that a complex number Yo, y=27 and plotting the function exp (2) upon a new plane, it follows at once from what has been said that every complex value =+in may have a limit it is necessary and sufficient that each of arises when s takes in turn all positions in this strip, and that of and 7 has a limit. Thus an infinite series wy+w1+w2+.. no value arises twice over. The equation exp (2) thus defines z, whose terms are complex numbers, is convergent if the real regarded as depending upon, with only an additive ambiguity series formed by taking the real parts of its terms and that 2ki, where k is an integer. We write z=A(); when is real this becomes the logarithm of ; in general X() = log is +i ph (5)+ formed by the imaginary terms are both convergent. The 2kzi, where k is an integer; and when describes a closed circuit series is also convergent if the real series formed by the moduli surrounding the origin the phase of increases by 2, or k increases of its terms is convergent; in that case the series is said to be by unity. Differentiating the series for we have dy/dz = 5, so absolutely convergent, and it can be shown that its sum is that 2, regarded as depending upon, is also differentiable, with unaltered by taking the terms in any other order. Generally-1)-...; it converges when =2 and hence converges for dz/dt = On the other hand, consider the series 5-1-4( − 1 )2 + the necessary and sufficient condition of convergence is that, for a given real positive e, a number m exists such that for every. n>m, and every positive p, the batch of terms w+wa++ +wa+p is less than e in absolute value. If the terms depend upon a complex variable z, the convergence is called uniform for a range of values of z, when the inequality holds, for the same e and m, for all the points z of this range.

2

-11; its differential coefficient is, however, 1-( − 1 ) + (-1)-.... that is, (1-1). Wherefore if (t) denote this series, for <1, the difference (5)-(5), regarded as a take the value of X() which vanishes when I we infer thence function of and has vanishing differential coefficients; if we -(-1). It is to be remarked that for 15-1|<1, A(5)='x (−1)′′-'{ that it is impossible for while subject to |-1|<1 to make a circuit about the origin. For values of for which -11, we can also calculate A() with the help of infinite series, utilizing the fact that λ(55') =λ(5) +λ(5′).

n=0

the form exp
the form exp (exp (2), with k=o, 1.....-1, all other

The infinite series of most importance are those of which the general term is a,z", wherein a, is a constant, and 2 is regarded as variable, n=0, 1, 2, 3,... Such a series is called a power series. If a real and positive number M exists such that for a=20 and every The function () is required to define when and a are complex n, 4.20" <M, a condition which is satisfied, for instance, if the numbers; this is defined as exp [ax(5) that is as "[\{})}*/n!. series converges for z=20, then it is at once proved that the series When a is a real integer the ambiguity of A() is immaterial here, converges absolutely for every z for which <o, and converges uniformly over every range || <r' for which '<where a is a positive integer, there are q values possible for e, of since exp [aλ(5)+2kari]=exp [ax()]; when a is of the form 1/g, To every power series there belongs then a circle of convergence within which it converges absolutely and uniformly; the function of z represented by it is thus continuous within the circle (this being the result of a general property of uniformly convergent series of continuous functions); the sum for an interior point is, however, continuous with the sum for a point on the circumference, as z approaches to zo provided the series converges for zzo, as can be shown without much difficulty. Within a common circle of convergence two power series Zanz, Eb, can be multiplied together according to the ordinary rule, this being a consequence of a theorem for absolutely convergent series. If r be less than the radius of convergence of a series Laz" and for |2|71, the sum of the series

values of k leading to one of these; the gth power of any one of
these values is ; when a plg, where p q are integers without
common factor, 9 being positive, we have (51/4)P. The
definition of the symbol is thus a generalization of the ordinary
definition of a power, when the numbers are real. As an example,
let it be required to find the meaning of ; the number i is of
modulus unity and phase }; thus A(i)=i(}x+2kx); thus
exp(-2kx) = exp (−1) exp (−2k),

is always real, but has an infinite number of values.

The function exp (2) is used also to define a generalized form of the cosine and sine functions when 2 is complex; we write, namely, cos 2=[exp (iz) + exp (−iz)] and sin z=i[exp (iz) —exp (−iz)]. It will be found that these obey the ordinary relations holding when z is real, except that their moduli are not inferior to unity. For example, cos i=1+1/2!+1/4!+...is obviously greater than unity. §4. Of Functions of a Complex Variable in General.—We have in what precedes shown how to generalize the ordinary rational, algebraic and logarithmic functions, and considered more general cases, of functions expressible by power series in s. With the suggestions furnished by these cases we can frame a general definition. So far our use of the plane upon which z is represented has been only illustrative, the results being capable of analytical statement. In what follows this representation is vital to the mode of expression we adopt; as then the properties of numbers cannot be ultimately based upon spatial intuitions, it is necessary to indicate what are the geometrical ideas requiring elucidation.

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definite finite real value attached to every interior or boundary point of the region, say f(x,y). It may have a finite upper limit H for the region, so that no point (x,y) exists for which f(x,y) > H, but points (x,y) exist for which f(x,y) > He, however small e may be; if not we say that its upper limit is infinite. There is then at least one point of the region such that, for points of the region within a circle about this point, the upper limit of f(x,y) is H, however small the radius of the circle be taken; for if not we can put about every point of the region a circle within which the upper limit of f(x,y) is less than H; then by the result (B) above the region consists of a finite number of sub-regions within each of which the upper limit is less than H; this is inconsistent with the hypothesis that the upper limit for the whole region is H. A similar statement holds for the lower limit. A case of such a function f(x,y) is the radius ro of the neighbourhood proper to any point 20, spoken of above. We can hence prove the statement (A) above. lower limit of ro is zero. Let then be a point such that the lower Suppose the property (2,2) extensive, and, if possible, that the limit of ro is zero for points zo within a circle about however small; let be the radius of the neighbourhood proper to ; take 20 so that -r; the property (2,0), being extensive, holds Consider a square of side a, to whose perimeter is attached a within a circle, centre zo, of radius -|-, which is greater definite direction of description, which we take to be counterthanzo-, and increases to as 20- diminishes; this being clockwise; another square, also of side a, may be added to this, so true for all points zo near 5, the lower limit of ro is not zero for the that there is a side common; this common side being erased we neighbourhood of, contrary to what was supposed. This proves have a composite region with a definite direction of perimeter: (A). Also, as is here shown that ror-o-, may similarly be to this a third square of the same size may be attached, so shown that rro-120-1. Thus ro differs arbitrarily little from that there is a side common to it and one of the former squares, ously with zo when zo is sufficiently small; that is, to varies continuand this common side may be erased. If this process be continued Next suppose the function f(x,y), which has a definite finite value at every point of the region considered, to be any number of times we obtain a region of the plane bounded by one continuous but not necessarily real, so that about every point ze. or more polygonal closed lines, no two of which intersect; and at each portion of the perimeter there is a definite direction of descrip- positive quantity assigned beforehand, a circle is possible, so that within or upon the boundary of the region, being an arbitrary real tion, which is such that the region is on the left of the describing for all points 2 of the region interior to this circle, we have point. Similarly we may construct a region by piecing together triangles, so that every consecutive two have a side in common, f(x,y)-(x,y)<, and therefore (x,y) being any other point it being understood that there is assigned an upper limit for the interior to this circle,|f(x,y)-f(x,y)|<n. We can then apply greatest side of a triangle, and a lower limit for the smallest angle. the result (A) obtained above, taking for the neighbourhood proper In the former method, each square may be divided into four others to any point 20 the circular arca within which, for any two points by lines through its centre parallel to its sides; in the latter method (x,y),(x,y), we have|f(x',y') − f(x,y)]<n. This is clearly an each triangle may be divided into four others by lines joining the extensive property. Thus, a number is assignable, greater than middle points of its sides;, this halves the sides and preserves the zero, such that, for any two points (x,y), (x,y) within a circle angles. When we speak of a region of the plane in general, unless about any point zo, we have f(x,y')—f(x,y) |<n. the contrary is stated, we shall suppose it capable of being generated in this latter way by means of a finite number of triangles, there being an upper limit to the length of a side of the triangle and a lower limit to the size of an angle of the triangle. We shall also require to speak of a path in the plane; this is to be understood as capable of arising as a limit of a polygonal path of finite length, there being a definite direction or sense of description at every point of the path, which therefore never meets itself. From this the meaning of a closed path is clear. The boundary points of a region form one or more closed paths, but, in general, it is only in a limiting sense that the interior points of a closed path are a region.

and, in particular, f(x,y)-f(x, y) In, where n is an arbitrary real positive quantity agreed upon beforehand.

Take now any path in the region, whose extreme points are e, 2, and let 1,... 2-1 be intermediate points of the path, in order; denote the continuous function f(x,y) by f(z), and let f, denote any quantity such that | fr-ƒ(2)| = |ƒ(~+1) — ƒ(2); consider the sum (220) fo+(-21)ƒ1+...+(2−Zn−1)ƒn-1. By the definition of a path we can suppose, n being large enough, that the intermediate points .... are so taken that if 2. zi, be any two points intermediate, in order, to z, and 2+1, we have 26+1-21 | < | 2+1 - &|; we can thus suppose | 1-20 | | 22-21 .... There is a logical principle also which must be referred to. We 2-2-1 fall to converge constantly to zero. This being so, we can frequently have cases where, about every, interior or boundary, show that the sum above has a definite limit. For this it is sufficient, point 2 of a certain region a circle can be put, say of radius ro, such as in the case of an integral of a function of one real variable, to that for all points of the region which are interior to this circle, prove this to be so when the convergence is obtained by taking new for which, that is, z-<ro, a certain property holds. Assuming points of division intermediate to the former ones. If, however, that to ro is given the value which is the upper limit for zo, of the 2r.1, 2.2.... Erm- be intermediate in order to 2 and 41, and possible values, we may call the points 12-20|<70, the neighbour-f-f(z) |<[ƒ(2,6+1)−ƒ(2,.) 1, the difference between (41)f hood belonging to or proper too, and may speak of the property and as the property (2,2). The value of ro will in general vary with zo; what is in most cases of importance is the question whether the lower limit of ro for all positions is zero or greater than zero. (A) This lower limit is certainly greater than zero provided the property (z,ze) is of a kind which we may call extensive; such, namely, that if it holds, for some position of and all positions of 2, within a certain region, then the property (2,2) holds within a circle of radius R about any interior point of this region for all points for which the circle 2-1-R is within the region. Also in this case ro varies continuously with 20. (B) Whether the property is of this extensive character or not we can prove that the region can be divided into a finite number of sub-regions such that, for every one of these, the property holds, (1) for some point zo within or upon the boundary of the sub-region, (2) for every point z within or upon the boundary of the sub-region.

We prove these statements (A), (B) in reverse order. To prove (B) let a region for which the property (2,20) holds for all points z and some point of the region, be called suitable: if each of the triangles of which the region is built up be suitable, what is desired is proved; if not let an unsuitable triangle be subdivided into four, as before explained; if one of these subdivisions is unsuitable let it be again subdivided; and so on. Either the process terminates and then what is required is proved; or else we obtain an indefinitely continued sequence of unsuitable trianglos, each contained in the preceding, which converge to a point, say ; after a certain stage all these will be interior to the proper region of ; this, however, is contrary to the supposition that they are all unsuitable.

We now make some applications of this result (B). Suppose a

{(2,1-2)fr.0+(2,2−27,1)fr.1+ . . . +(2x+1—2r,m−1)fr,m−1}

which is equal to

is, when 1-2 is small enough, to ensure [ƒ(2+1) −ƒ(2) ] < n, less in absolute value than

2272|2,1+1=2,i |, which, if S be the upper limit of the perimeter of the polygon from which the path is generated, is < 27S, and is therefore arbitrarily small.

The limit in question is called f, f(2)dz. In particular when f(z) =1, it is obvious from the definition that its value is 2-20: when f(s), by taking f,= (1-2), it is equally clear that its value is (2-2); these results will be applied immediately.

certain region there belong two definite finite numbers f(20), F(20), Suppose now that to every interior and boundary point 2 of a such that, whatever real positive quantity may be, a real positive numbere exists for which the condition