Chapter I: Functions of Real Variables (6)

these quantities cannot therefore have a real ratio, for else, being
periods of a monogenic function, they would, as we have previously
seen, be each integral multiples of another period; there would then
be a closed path for (x, y), starting from an arbitrary point (x0,
y0), other than one enclosing two of the points (e1, 0), (e2, 0), (e3,
0), ([oo], [oo]), which leads back to the initial point (x0, y0),
which is impossible. On the whole, therefore, it appears that the
function [phi](u) agrees with the function RN(u) previously discussed,
and the discussion of the elliptic integrals can be continued in the
manner given under S 14, _Doubly Periodic Functions_.

S 21. _Modular Functions._--One result of the previous theory is the
remarkable fact that if
_ _
/ [oo] dx / [oo] dx
[omega] = 2| --, [omega]' = 2 | --,
_/ e1 y _/ e3 y

where y^2 = 4(x - e1) (x - e2) (x - e3), then we have

e1 = (1/2[omega])^(-2) + [Sigma]' {[(m + 1/2)[omega] + m'[omega]']^(-2) -
[m[omega] + m'[omega]']^(-2)},

and a similar equation for e3, where the summation refers to all integer values of m and m' other than the one pair m = 0, m' = 0. This, with similar results, has led to the consideration of functions of the complex ratio [omega]'/[omega].

It is easy to see that the series for RN(u), u^(-2) + [Sigma] [(u +
m[omega] + m'[omega]')^2-(m[omega] + m'[omega]')^2], is unaffected by
replacing [omega], [omega]' by two quantities [Omega], [Omega]' equal
respectively to p[omega] + q[omega]', p'[omega]' + q'[omega]', where
p, q, p', q' are any integers for which pq' - p'q = [+-]1; further it
can be proved that all substitutions with integer coefficients [Omega]
= p[omega] + q[omega]', [Omega]' = p'[omega] + q'[omega]', wherein pq'
- p'q = 1, can be built up by repetitions of the two particular
substitutions ([Omega] = -[omega]', [Omega]' = [omega]), ([Omega] =
[omega], [Omega]' = [omega] + [omega]'). Consider the function of the
ratio [omega]'/[omega] expressed by

h = -RN (1/2[omega]') / RN(1/2[omega]);

it is at once seen from the properties of the function RN(u) that by
the two particular substitutions referred to we obtain the
corresponding substitutions for h expressed by

h' = 1/h, h' = 1 - h;

thus, by all the integer substitutions [Omega] = p[omega] + q[omega]',
[Omega]' = p'[omega] + q'[omega]', in which pq' - p'q = 1, the
function h can only take one of the six values h, 1/h, 1 - h, 1/(1 -
h), h/(h - 1), (h - 1)/h, which are the roots of an equation in
[theta],

(1 - [theta] + [theta]^2)^3 (1 - h + h^2)^3
--------------------------- = ---------------;
[theta]^2(1 - [theta])^2 h^2(1 - h)^2

the function of [tau], = [omega]'/[omega], expressed by the right
side, is thus unaltered by every one of the substitutions [tau]' = (p'
+ q'[tau] / p + q[tau]), wherein p, q, p', q' are integers having pq'
- p'q = 1. If the imaginary part [sigma], of [tau], which we may write
[tau] = [rho] + i[sigma], is positive, the imaginary part of [tau]',
which is equal to [sigma](pq' - p'q)/[(p + q[rho])^2 + q^2[sigma]^2],
is also positive; suppose [sigma] to be positive; it can be shown that
the upper half of the infinite plane of the complex variable [tau] can
be divided into regions, all bounded by arcs of circles (or straight
lines), no two of these regions overlapping, such that any
substitution of the kind under consideration, [tau]' = (p' +
q'[tau])/(p + q[tau]) leads from an arbitrary point [tau], of one of
these regions, to a point [tau]' of another; taking [tau] = [rho] +
i[sigma], one of these regions may be taken to be that for which -1/2
< [rho] < 1/2, [rho]^2 + [sigma]^2 > 1, together with the points for
which [rho] is negative on the curves limiting this region; then every
other region is obtained from this so-called fundamental region by one
and only one of the substitutions [tau] = (p' + q'[tau])/(p + q[tau]),
and hence by a definite combination of the substitutions [tau]' =
-1/[tau], [tau]' = 1 + [tau]. Upon the infinite half plane of [tau],
the function considered above,

4 [RN^2(1/2[omega]) + RN(z(1/2[omega]) RN(1/2[omega]') + RN^2 (1/2[omega]')]^3
z([tau]) = -- ----------------------------------------------------------------------------
27 RN^2(1/2[omega]) RN^2(1/2[omega]') [RN(1/2[omega]) + RN(1/2[omega]')]^2

is a single valued monogenic function, whose only essential
singularities are the points [tau]' = (p' + q'[tau])/(p + q[tau]) for
which [tau] = [oo], namely those for which [tau]' is any real rational
value; the real axis is thus a line over which the function z([tau])
cannot be continued, having an essential singularity in every arc of
it, however short; in the fundamental region, z([tau]) has thus only
the single essential singularity, r = [rho] + i[sigma], where [sigma]
= [oo]; in this fundamental region z([tau]) takes any assigned complex
value just once, the relation z([tau]') = z([tau]) requiring, as can
be shown, that [tau]' is of the form (p' + q'[tau])/(p + q[tau]), in
which p, q, p', q' are integers with pq' - p'q = 1; the function
z([tau]) has thus a similar behaviour in every other of the regions.
The division of the plane into regions is analogous to the division of
the plane, in the case of doubly periodic functions, into
parallelograms; in that case we considered only functions without
essential singularities, and in each of the regions the function
assumed every complex value twice, at least. Putting, as another
function of [tau], J([tau]) = z([tau])[z([tau]) - 1], it can be shown
that J([tau]) = 0 for [tau] = exp (2/3[pi]i), that J([tau]) = 1 for
[tau] = i, these being values of [tau] on the boundary of the
fundamental region; like z([tau]) it has an essential singularity for
[tau] = [rho] + i[sigma], [sigma] = + [oo]. In the theory of linear
differential equations it is important to consider the inverse
function [tau](J); this is infinitely many valued, having a cycle of
three values for circulation of J about J = 0 (the circuit of this
point leading to a linear substitution for [tau] of period 3, such as
[tau]' = -(1 + [tau])^(-1)), having a cycle of two values about J = 1
(the circuit leading to a linear substitution for [tau] of period 2,
such as [tau]' = -[tau]^(-1)), and having a cycle of infinitely many
values about J = [oo] (the circuit leading to a linear substitution
for [tau] which is not periodic, such as [tau]' = 1 + [tau]). These
are the only singularities for the function [tau](J). Each of the
functions
_ _
| RN(1/2[omega]) + 2RN(1/2[omega]') |^(1/8)
[J([tau])]^(1/3), [J([tau])-1]^1/2, | - --------------------------------- | ,
|_ RN(1/2[omega]) - RN(1/2)[omega]') _|

beside many others (see below), is a single valued function of [tau],
and is expressible without ambiguity in terms of the single valued
function of [tau],

/i[pi][tau]\ [oo]
[eta]([tau]) = exp( ---------- ) [Pi] [1-exp (2i[pi]n[tau])],
\ 12 / n=1

/i[pi][tau]\ _[oo]
= exp(----------- ) \ (-1)^m exp [(3m^2 + m) i[pi][tau]].
\ 12 / /_
m = -[oo]

It should be remarked, however, that [eta]([tau]) is not unaltered by
all the substitutions we have considered; in fact

[eta](-[tau]^(-1)) = (-i[tau])1/2[eta]([tau]), [eta](1 + [tau]) =
exp (1/12 i[pi]) [eta]([tau]).

The aggregate of the substitutions [tau]' = (p' + q'[tau])/(p +
q[tau]), wherein p, q, p', q' are integers with pq' - p'q = 1,
represents a _Group_; the function J([tau]), unaltered by all these
substitutions, is called a _Modular Function_. More generally any
function unaltered by all the substitutions of a group of linear
substitutions of its variable is called an _Automorphic Function_. A
rational function, of its variable h, of this character, is the
function (1 - h + h^2)^3 h^(-2)(1 - h)^(-2) presenting itself
incidentally above; and there are other rational functions with a
similar property, the group of substitutions belonging to any one of
these being, what is a very curious fact, associable with that of the
rotations of one of the regular solids, about an axis through its
centre, which bring the solid into coincidence with itself. Other
automorphic functions are the double periodic functions already
discussed; these, as we have seen, enable us to solve the algebraic
equation y^2 = 4x^3 - g2x - g3 (and in fact many other algebraic
equations, see below, under S 23, _Geometrical Applications of
Elliptic Functions_) in terms of single valued functions x = RN(u), y
= -RN'(u). A similar utility, of a more extended kind, belongs to
automorphic functions in general; but it can be shown that such
functions necessarily have an infinite number of essential
singularities except for the simplest cases.

The modular function J([tau]) considered above, unaltered by the group
of linear substitutions [tau]' = (p' + q'[tau]) / (p + q[tau]), where
p, q, p', q' are integers with pq' - p'q = 1, may be taken as the
independent variable x of a differential equation of the third order,
of the form

s''' 3 /s''\^2 1 - a^2 1 - [beta]^2 [alpha]^2 + [beta]^2 - [gamma]^2 - 1
---- - -- ( --- ) = ---------- + ------------ + ------------------------------------,
s' 2 \ s'/ 2(x - 1)^2 2x^2 2x(x - 1)

where s' = ds/dx, &c., of which the dependent variable s is equal to
[tau]. A differential equation of this form is satisfied by the
quotient of two independent integrals of the linear differential
equation of the second order satisfied by the hypergeometric
functions. If the solution of the differential equation for s be
written s([alpha], [beta], [gamma], x), we have in fact [tau] = s(1/2,
1/3, 0, J). If we introduce also the function of [tau] given by

2RN (1/2[omega]') + f V(1/2[omega])
[lambda] = -----------------------------------,
RN(1/2[omega]') - RN(1/2[omega])

we similarly have [tau] = s(0, 0, 0, [lambda]); this function [lambda]
is a single valued function of [tau], which is also a modular
function, being unaltered by a group of integral substitutions also of
the form [tau]' = (p' + q'[tau])/(p + q[tau]), with pq' - p'q = 1, but
with the restriction that p' and q are even integers, and therefore p
and q' are odd integers. This group is thus a subgroup of the general
modular group, and is in fact of the kind called a self-conjugate
subgroup. As in the general case this subgroup is associated with a
subdivision of the plane into regions of which any one is obtained
from a particular region, called the fundamental region, by a
particular one of the substitutions of the subgroup. This fundamental
region, putting [tau] = [rho] + i[sigma], may be taken to be that
given by -1 < [rho] < 1, ([rho] + 1/2)^2 + [sigma]^2 > 1/4, ([rho] -
1/2)^2 + [sigma]^2 > 1/4, and is built up of six of the regions which
arose for the general modular group associated with J([tau]). Within
this fundamental region, [lambda] takes every complex value just once,
except the values [lambda] = 0, 1, [oo], which arise only at the
angular points [tau] = 0, [tau] = [oo], [tau] = -1 and the equivalent
point [tau] = 1; these angular points are essential singularities for
the function [lambda]([tau]). For [lambda]([tau]) as for J([tau]), the
region of existence is the upper half plane of [tau], there being an
essential singularity in every length of the real axis, however short.

If, beside the plane of [tau], we take a plane to represent the values
of [lambda], the function [tau] = s(0, 0, 0, [lambda]) being
considered thereon, the values of [tau] belonging to the interior of
the fundamental region of the [tau]-plane considered above, will
require the consideration of the whole of the [lambda]-plane taken
once with the exception of the portions of the real axis lying between
-[oo] and 0 and between 1 and + [oo], the two sides of the first
portion corresponding to the circumferences of the [tau]-plane
expressed by ([rho] + 1/2)^2 + [sigma]^2 = 1/4, ([rho] - 1/2)^2 +
[sigma]^2 = 1/4, while the two sides of the latter portion, for which
[lambda] is real and > 1, correspond to the lines of the [tau]-plane
expressed by [rho] = [+-]1. The line for which [lambda] is real,
positive and less than unity corresponds to the imaginary axis of the
[tau]-plane, lying in the interior of the fundamental region. All the
values of [tau] = s(0, 0, 0, [lambda]) may then be derived from those
belonging to the fundamental region of the [tau]-plane by making
[lambda] describe a proper succession of circuits about the points
[lambda] = 0, [lambda] = 1; any such circuit subjects [tau] to a
linear substitution of the subgroup of [tau] considered, and
corresponds to a change of [tau] from a point of the fundamental
region to a corresponding point of one of the other regions.

S 22. _A Property of Integral Functions deduced from the Theory of Modular Functions._--Consider now the function exp(z), for finite values of z; for such values of z, exp(z) never vanishes, and it is impossible to assign a closed circuit for z in the finite part of the plane of z which will make the function [lambda] = exp(z) pass through a closed succession of values in the plane of [lambda] having [lambda] = 0 in its interior; the function s[0, 0, 0, exp(z)], however z vary in the finite part of the plane, will therefore never be subjected to those linear substitutions imposed upon s(0, 0, 0, [lambda]) by a circuit of [lambda] about [lambda] = 0; more generally, if [phi](z) be an integral function of z, never becoming either zero or unity for finite values of z, the function [lambda] = [phi](z), however z vary in the finite part of the plane, will never make, in the plane of [lambda], a circuit about either [lambda] = 0 or [lambda] = 1, and s(0, 0, 0, [lambda]), that is s[0, 0, 0, [phi](z)], will be single valued for all finite values of z; it will moreover remain finite, and be monogenic. In other words, s[0, 0, 0, [phi](z)] is also an integral function--whose imaginary part, moreover, by the property of s(0, 0, 0, [lambda]), remains positive for all finite values of z. In that case, however, exp{is[0, 0, 0, [phi](z)]} would also be an integral function of z with modulus less than unity for all finite values of z. If, however, we describe a circle of radius R in the z plane, and consider the greatest value of the modulus of an integral function upon this circle, this certainly increases indefinitely as R increases. We can infer therefore that _an integral function [phi](z) which does not vanish for any finite value of z, takes the value unity and hence_ (by considering the function A^(-1)[phi](z)) _takes every other value for some definite value of z_; or, an integral function for which both the equations [phi](z) = A, [phi](z) = B are unsatisfied by definite values of z, does not exist, A and B being arbitrary constants.

A similar theorem can be proved in regard to the values assumed by the
function [phi](z) for points z of modulus greater than R, however
great R may be, also with the help of modular functions. In general
terms it may be stated that it is a very exceptional thing for an
integral function not to assume every complex value an infinite number
of times.

Another application of modular functions is to prove that the function
s([alpha], [beta], [gamma], [lambda]) is a single valued function of
[tau] = s(0, 0, 0, [lambda]); for, putting [tau]' = ([tau] - i)/([tau]
+ i), the values of [tau]' which correspond to the singular points
[lambda] = 0, 1, [oo] of s([alpha], [beta], [gamma], [lambda]), though
infinite in number, all lie on the circumference of the circle
|[tau]'| = 1, within which therefore s([alpha], [beta], [gamma], x) is
expressible in a form [Sigma] [n = 0 to [oo]] a_n[tau]'^n. More
generally any monogenic function of [lambda] which is single valued
save for circuits of the points [lambda] = 0, 1, [oo], is a single
valued function of [tau] = s(0, 0, 0, [lambda]). Identifying [lambda]
with the square of the modulus in Legendre's form of the elliptical
integral, we have [tau] = iK'/K, where

_ _
/1 dt /1 dt
K = | ----------------------------------, K' = | ---------------------------------------;
_/0 [root][1 - t^2] [1 - [lambda]t^2] _/0 [root][1 - t^2] [1 - (1 - [lambda])t^2]

functions such as [lambda]^1/4, (1 - [lambda])^1/4, [[lambda](1 -
[lambda])]^1/4, which have only [lambda] = 0, 1, [oo] as singular
points, were expressed by Jacobi as power series in q =
e^(i[pi][tau]), and therefore, at least for a limited range of values
of [tau], as single valued functions of [tau]; it follows by the
theorem given that any product of a root of [lambda] and a root of 1 -
[lambda] is a single valued function of [tau]. More generally the
differential equation

d^2y dy
x(1 - x) ---- + [[gamma] - ([alpha] + [beta] + 1)x] -- -[alpha][beta][gamma] = 0
dx^2 dx

may be solved by expressing both the independent and dependent
variables as single valued functions of a single variable [tau], the
expression for the independent variable being x = [lambda]([tau]).

S 23. _Geometrical Applications of Elliptic Functions._--Consider any irreducible algebraic equation rational in x, y, f(x, y) = 0, of such a form that the equation represents a plane curve of order n with 1/2n(n -3) double points; taking upon this curve n-3 arbitrary fixed points, draw through these and the double points the most general curve of order n -2; this will intersect [f] in n(n - 2) - n(n - 3) - (n - 3) = 3 other points, and will contain homogeneously at least 1/2(n - 1)n - 1/2n(n -3) - (n - 3) = 3 arbitrary constants, and so will be of the form [lambda][phi] + [lambda]1[phi]1 + [lambda]2[phi]2 + ... = 0, wherein [lambda]3, [lambda]4, ... are in general zero. Put now [xi] = [phi]1/[phi], [eta] = [phi]2/[phi] and eliminate x, y between these equations and [f](x, y) = 0, so obtaining a rational irreducible equation F([xi], [eta]) = 0, representing a further plane curve. To any point (x, y) of [f] will then correspond a definite point ([xi], [eta]) of F.

For a general position of (x, y) upon [f] the equations [phi]1(x',
y')/[phi](x', y') = [phi]1(x, y)/[phi](x, y), [phi]2(x', y')/[phi](x',
y') = [phi]2(x, y)/[phi](x, y), subject to [f](x', y') = 0, will have
the same number of solutions (x', y'); if their only solution is x' =
x, y' = y, then to any position ([xi],[eta]) of F will conversely
correspond only one position (x, y) of [f]. If these equations have
another solution beside (x, y), then any curve [lambda][phi] +
[lambda]1[phi]1 + [lambda]2[phi]2 = 0 which passes (through the double
points of [f] and) through the n - 2 points of [f] constituted by the
fixed n-3 points and a point (x0, y0), will necessarily pass through a
further point, say (x0', y0'), and will have only one further
intersection with [f]; such a curve, with the n - 2 assigned points,
beside the double points, of [f], will be of the form [mu][psi] +
[mu]1[psi]1 + ... = 0, where [mu]2, [mu]3, ... are generally zero;
considering the curves [psi] + t[psi]1 = 0, for variable t, one of
these passes through a further arbitrary point of [f], by choosing t
properly, and conversely an arbitrary value of t determines a single
further point of [f]; the co-ordinates of the points of [f] are thus
rational functions of a parameter t, which is itself expressible
rationally by the co-ordinates of the point; it can be shown
algebraically that such a curve has not 1/2(n - 3)n but 1/2(n - 3)n +
1 double points. We may therefore assume that to every point of F
corresponds only one point of [f], and there is a birational
transformation between these curves; the coefficients in this
transformation will involve rationally the co-ordinates of the n-3
fixed points taken upon [f], that is, at the least, by taking these to
be consecutive points, will involve the co-ordinates of one point of
[f], and will not be rational in the coefficients of [f] unless we can
specify a point of [f] whose co-ordinates are rational in these. The
curve F is intersected by a straight line a[xi] + b[eta] + c = 0 in as
many points as the number of unspecified intersections of [f] with
a[phi] + b[phi]1 + c[phi]2 = 0, that is, 3; or F will be a cubic
curve, without double points.

Such a cubic curve has at least one point of inflection Y, and if a
variable line YPQ be drawn through Y to cut the curve again in P and
Q, the locus of a point R such that YR is the harmonic mean of YP and
YQ, is easily proved to be a straight line. Take now a triangle of
reference for homogeneous co-ordinates XYZ, of which this straight
line is Y = 0, and the inflexional tangent at Y is Z = 0; the equation
of the cubic curve will then be of the form

ZY^2 = aX^3 + bX^2Z + cXZ^2 + dZ^3;

by putting X equal to [lambda]X + [mu]Z, that is, choosing a suitable
line through Y to be X = 0, and choosing [lambda] properly, this is
reduced to the form

ZY^2 = 4X^3 - g2XZ^2 -g3Z^3,

of which a representation is given, valid for every point, in terms of
the elliptic functions RN(u), RN'(u), by taking X = ZRN(u), Y =
ZRN'(u). The value of u belonging to any point is definite save for
sums of integral multiples of the periods of the elliptic functions,
being given by
_
/ (x) ZdX - XdZ
u = | ---------,
_/ ([oo]) ZY

where ([oo]) denotes the point of inflection.

It thus appears that the co-ordinates of any point of a plane curve,
[f], of order n with 1/2(n - 3)n double points are expressible as
elliptic functions, there being, save for periods, a definite value of
the argument u belonging to every point of the curve. It can then be
shown that if a variable curve, [phi], of order m be drawn, passing
through the double points of the curve, the values of the argument u
at the remaining intersections of [phi] with [f], have a sum which is
unaffected by variation of the coefficients of [phi], save for
additive aggregates of the periods. In virtue of the birational
transformation this theorem can be deduced from the theorem that if
any straight line cut the cubic y^2 = 4x^3 - g2x - g3, in points (u1),
(u2), (u3), the sum u1 + u2 + u3 is zero, or a period; or the general
theorem is a corollary from Abel's theorem proved under S 17,
_Integrals of Algebraic Functions_. To prove the result directly for
the cubic we remark that the variation of one of the intersections (x,
y) of the cubic with the straight line y = mx + n, due to a variation
[delta]m, [delta]n in m and n, is obtained by differentiation of the
equation for the three abscissae, namely the equation

F(x) = 4x^3 - g2x - g3 - (mx + n)^2 = 0,

and is thus given by

dx x[delta]m + [delta]n
-- = 2 --------------------,
y F'(x)

and the sum of three such fractions as that on the right for the three
roots of F(x) = 0 is zero; hence u1 + u2 + u3 is independent of the
straight line considered; if in particular this become the inflexional
tangent each of u1, u2, u3 vanishes. It may be remarked in passing
that x1 + x2 + x3 = 1/4m^2, and hence is 1/4{(y1 - y2)/(x1 - x2)}^2; so
that we have another proof of the addition equation for the function
RN(u). From this theorem for the cubic curve many of its geometrical
properties, as for example those of its inflections, the properties of
inscribed polygons, of the three kinds of corresponding points, and
the theory of residuation, are at once obvious. And similar results
hold for the curve of order n with 1/2(n - 3)n double points.

S 24. _Integrals of Algebraic Functions in Connexion with the Theory of Plane Curves._--The developments which have been explained in connexion with elliptic functions may enable the reader to appreciate the vastly more extensive theory similarly arising for any algebraical irrationality, [f](x, y) = o.

The algebraical integrals [int] R(x, y)dx associated with this may as
before be divided into those of the _first kind_, which have no
infinities, those of the _second kind_, possessing only algebraical
infinities, and those of the _third kind_, for which logarithmic
infinities enter. Here there is a certain number, p, greater than
unity, of linearly independent integrals of the first kind; and this
number p is unaltered by any birational transformation of the
fundamental equation [f](x, y) = 0; a rational function can be
constructed with poles of the first order at p + 1 arbitrary positions
(x, y), satisfying [f](x, y) = 0, but not with a fewer number unless
their positions are chosen properly, a property we found for the case
p = 1; and p is the number of linearly independent curves of order n-3
passing through the double points of the curve of order n expressed by
[f](x, y) = 0. Again any integral of the second kind can be expressed
as a sum of p integrals of this kind, with poles of the first order at
arbitrary positions, together with rational functions and integrals of
the first kind; and an integral of the second kind can be found with
one pole of the first order of arbitrary position, and an integral of
the third kind with two logarithmic infinities, also of arbitrary
position; the corresponding properties for p = 1 are proved above.

There is, however, a difference of essential kind in regard to the
inversion of integrals of the first kind; if u = [int] R(x, y)dx be
such an integral, it can be shown, in common with all algebraic
integrals associated with [f](x, y) = 0, to have 2p linearly
independent additive constants of indeterminateness; the upper limit
of the integral cannot therefore, as we have shown, be a single valued
function of the value of the integral. The corresponding theorem, if
[int] R_i(x, y)dx denote one of the integrals of the first kind, is
that the p equations
_ _
/ /
| R_i (x1, y1)dx1 + ... + | Ri (x_p, y_p) dx_p = u_i,
_/ _/

determine the rational symmetric functions of the p positions (x1,
y1), ... (x_p, y_p) as single valued functions of the p variables, u1,
... u_p. It is thus necessary to enter into the theory of functions of
several independent variables; and the equation [f](x, y) = 0 is thus
not, in this way, capable of solution by single valued functions of
one variable. That solution in fact is to be sought with the help of
automorphic functions, which, however, as has been remarked, have, for
p > 1, an infinite number of essential singularities.

S 25. _Monogenic Functions of Several Independent Variables._--A monogenic function of several independent complex variables u_i, ... u_p is to be regarded as given by an aggregate of power series all obtainable by continuation from any one of them in a manner analogous to that before explained in the case of one independent variable. The singular points, defined as the limiting points of the range over which such continuation is possible, may either be _poles_, or _polar points of indetermination_, or _essential singularities_.

A pole is a point (u1^(0), ... u_p^(0)) in the neighbourhood of which
the function is expressible as a quotient of converging power series
in u1 - u1^(0) ... u_p - u_p^(0); of these the denominator series D
must vanish at (u1^(0), ... u_p^(0)), since else the fraction is
expressible as a power series and the point is not a singular point,
but the numerator series N must not also vanish at (u1^(0), ...
u_p^(0)), or if it does, it must be possible to write D = M0, N = MN0,
where M is a converging power series vanishing at (u1^(0), ...
u_p^(0)), and N0 is a converging power series, in (u1 - u1^(0) ... u_p
- u_p^(0)), not so vanishing. A polar point of indetermination is a
point about which the function can be expressed as a quotient of two
converging power series, both of which vanish at the point. As in such
a simple case as (Ax + By) / (ax + by), about x = 0, y = 0, it can be
proved that then the function can be made to approach to any
arbitrarily assigned value by making the variables u1, ... u_p
approach to u1^(0), ... u_p^(0) by a proper path. It is the necessary
existence of such polar points of indetermination, which in case p > 2
are not merely isolated points, which renders the theory essentially
more difficult than that of functions of one variable. An essential
singularity is any which does not come under one of the two former
descriptions and includes very various possibilities. A point at
infinity in this theory is one for which any one of the variables u1,
... u_p is indefinitely great; such points are brought under the
preceding definitions by means of the convention that for u_i^(0) =
[oo], the difference u_i - u_i^(0) is to be understood to stand for
u_i^(-1) . This being so, a single valued function of u1, ... u_p
without essential singularities for infinite or finite values of the
variables can be shown, by induction, to be, as in the case of p = 1,
necessarily a rational function of the variables. A function having no
singularities for finite values of all the variables is as before
called an integral function; it is expressible by a power series
converging for all finite values of the variables; a single valued
function having for finite values of the variables no singularities
other than poles or polar points of indetermination is called a
meromorphic function; as for p = 1 such a function can be expressed as
a quotient of two integral functions having no common zero point other
than the points of indetermination of the function; but the proof of
this theorem is difficult.

The single valued functions which occur, as explained above, in the
inversion of algebraic integrals of the first kind, for p > 1, are
meromorphic. They must also be periodic, unaffected that is when the
variables u1, ... u_p are _simultaneously_ increased each by a proper
constant, these being the additive constants of indeterminateness for
the p integrals [int] R_i(x, y)dx arising when (x, y) makes a closed
circuit, the same for each integral. The theory of such single valued
meromorphic periodic functions is simpler than that of meromorphic
functions of several variables in general, as it is sufficient to
consider only finite values of the variables; it is the natural
extension of the theory of doubly periodic functions previously
discussed. It can be shown to reduce, though the proof of this
requires considerable developments of which we cannot speak, to the
theory of a single integral function of u1, ... u_p, called the _Theta
Function_. This is expressible as a series of positive and negative
integral powers of quantities exp (c1u1), exp (c2u2), ... exp (c_p
u_p), wherein c1, ... c_p are proper constants; for p = 1 this theta
function is essentially the same as that above given under a different
form (see S 14, _Doubly Periodic Functions_), the function [sigma](u).
In the case of p = 1, all meromorphic functions periodic with the same
two periods have been shown to be rational functions of two of them
connected by a single algebraic equation; in the same way all
meromorphic functions of p variables, periodic with the same sets of
simultaneous periods, 2p sets in all, can be shown to be expressible
rationally in terms of p + 1 such periodic functions connected by a
single algebraic equation. Let x1, ... x_p, y denote p + 1 such
functions; then each of the partial derivatives dx_i/(Pd)u_i will
equally be a meromorphic function of the same periods, and so
expressible rationally in terms of x1, ... x_p, y; thus there will
exist p equations of the form

dx_i = R1 du1 + ... + R_p du_p,

and hence p equations of the form

du_i = H_(i, 1)dx1 + ... + H_(i, p)dx_p,

wherein H_(i, j) are rational functions of x1, ... x_p, y, these being
connected by a fundamental algebraic (rational) equation, say [f](x1,
... x_p, y) = 0. This then is the generalized form of the
corresponding equation for p = 1.

S 26. _Multiply-Periodic Functions and the Theory of Surfaces._--The theory of algebraic integrals [int] R(x, y)dx, wherein x, y are connected by a rational equation [f](x, y) = 0, has developed concurrently with the theory of algebraic curves; in particular the existence of the number p invariant by all birational transformations is one result of an extensive theory in which curves capable of birational correspondence are regarded as equivalent; this point of view has made possible a general theory of what might otherwise have remained a collection of isolated theorems.

In recent years developments have been made which point to a similar
unity of conception as possible for surfaces, or indeed for algebraic
constructs of any number of dimensions. These developments have been
in two directions, at first followed independently, but now happily
brought into the most intimate connexion. On the analytical side, E.
Picard has considered the possibility of classifying integrals of the
form [int](Rds + Sdy), belonging to a surface [f](x, y, z) = 0,
wherein R and S are rational functions of x, y, z, according as they
are (1) everywhere finite, (2) have poles, which then lie along curves
upon the surface, or (3) have logarithmic infinities, also then lying
along curves, and has brought the theory to a high degree of
perfection. On the geometrical side A. Clebsch and M. Noether, and
more recently the Italian school, have considered the geometrical
characteristics of a surface which are unaltered by birational
transformation. It was first remarked that for surfaces of order n
there are associated surfaces of order n-4, having properties in
relation thereto analogous to those of curves of order n-3 for a plane
curve of order n; if such a surface [f](x, y, z) = 0 have a double
curve with triple points triple also for the surface, and [phi](x, y,
z) = 0 be a surface of order n - 4 passing through the double curve,
the double integral
_ _
/ / [phi] dx dy
| | -----------
_/ _/ (Pd)f/(Pd)z

is everywhere finite; and, the most general everywhere finite integral
of this form remains invariant in a birational transformation of the
surface [f], the theorem being capable of generalization to algebraic
constructs of any number of dimensions. The number of linearly
independent surfaces of order n - 4, possessing the requisite
particularity in regard to the singular lines and points of the
surface, is thus a number invariant by birational transformation, and
the equality of these numbers for two surfaces is a necessary
condition of their being capable of such transformation. The number of
surfaces of order m having the assigned particularity in regard to the
singular points and lines of the fundamental surface can be given by a
formula for a surface of given singularity; but the value of this
formula for m = n - 4 is not in all cases equal to the actual number
of surfaces of order n - 4 with the assigned particularity, and for a
cone (or ruled surface) is in fact negative, being the negative of the
deficiency of the plane section of the cone. Nevertheless this number
for m = n - 4 is also found to be invariant for birational
transformation. This number, now denoted by p_a, is then a second
invariant of birational transformation. The former number, of actual
surfaces of order n - 4 with the assigned particularity in regard to
the singularities of the surface, is now denoted by p_g. The
difference p_g - p_a, which is never negative, is a most important
characteristic of a surface. When it is zero, as in the case of the
general surface of order n, and in a vast number of other ordinary
cases, the surface is called regular.

On a plane algebraical curve we may consider linear series of sets of
points, obtained by the intersection with it of curves [lambda][phi] +
[lambda]1[phi]1 + ... = 0, wherein [lambda], [lambda]1, ... are
variable coefficients; such a series consists of the sets of points
where a rational function of given poles, belonging to the construct
[f](x, y) = 0, has constant values. And we may consider series of sets
of points determined by variable curves whose coefficients are
algebraical functions, not necessarily rational functions, of
parameters. Similarly on a surface we may consider linear systems of
curves, obtained by the intersection with the given surface of
variable surfaces [lambda][phi] + [lambda]1[phi]1 + ... = 0, and may
consider algebraic systems, of which the individual curve is given by
variable surfaces whose coefficients are algebraical, not necessarily
rational, functions of parameters. Of a linear series upon a plane
curve there are two numbers manifestly invariant in birational
transformation, the _order_, which is the number of points forming a
set of the series, and the _dimension_, which is the number of
parameters [lambda]1/[lambda], [lambda]2/[lambda], ... entering
linearly in the equation of the series. The series is _complete_ when
it is not contained in a series of the same order but of higher
dimension. So for a linear system of curves upon a surface, we have
three invariants for birational transformation; the _order_, being in
the number of variable intersections of two curves of the system, the
_dimension_, being the number of linear parameters [lambda]1/[lambda],
[lambda]2/[lambda], ... in the equation for the system, and the
_deficiency_ of the individual curves of the system. Upon any curve of
the linear system the other curves of the system define a linear
series, called the _characteristic_ series; but even when the linear
system is complete, that is, not contained in another linear system of
the same order and higher dimension, it does not follow that the
characteristic series is complete; it may be contained in a series
whose dimension is greater by p_g - p_a than its own dimension. When
this is so it can be shown that the linear system of curves is
contained in an algebraic system whose dimension is greater by p_g -
p_a than the dimension of the linear system. The extra p = p_g - p_a
variable parameters so entering may be regarded as the independent
co-ordinates of an algebraic construct [f](y, x1, ... x_p) = 0; this
construct has the property that its co-ordinates are single valued
meromorphic functions of p variables, which are periodic, possessing
2p systems of periods; the p variables are expressible in the forms
_
/
u_i = | R1(x, y) dx1 + ... + R_p(x, y) dx_p,
_/

wherein R_i(x, y) denotes a rational function of x1, ... x_p and y.
The original surface has correspondingly p integrals of the form
[int](R dx + S dy), wherein R, S are rational in x, y, z, which are
everywhere finite; and it can be shown that it has no other such
integrals. From this point of view, then, the number p, = p_g - p_a
is, for a surface, analogous to the deficiency of a plane curve;
another analogy arises in the comparison of the theorems: for a plane
curve of zero deficiency there exists no algebraic series of sets of
points which does not consist of sets belonging to a linear series;
for a surface for which p_g - p_a = 0 there exists no algebraic system
of curves not contained in a linear system.

But whereas for a plane curve of deficiency zero, the co-ordinates of
the points of the curve are rational functions of a single parameter,
it is not necessarily the case that for a surface having p_g - p_a = 0
the co-ordinates of the points are rational functions of two
parameters; it is necessary that p_g - p_a = 0, but this is not
sufficient. For surfaces, beside the p_g linearly independent surfaces
of order n - 4 having a definite particularity at the singularities of
the surface, it is useful to consider surfaces of order k(n - 4), also
having each a definite particularity at the singularities, the number
of these, not containing the original surface as component, which are
linearly independent, is denoted by P_k. It can then be stated that a
sufficient condition for a surface to be rational consists of the two
conditions p_a = 0, P2 = 0. More generally it becomes a problem to
classify surfaces according to the values of the various numbers which
are invariant under birational transformation, and to determine for
each the simplest form of surface to which it is birationally
equivalent. Thus, for example, the hyperelliptic surface discussed by
Humbert, of which the co-ordinates are meromorphic functions of two
variables of the simplest kind, with four sets of periods, is
characterized by p_g = 1, p_a = -1; 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 mathematique_, 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. Meray, _Lecons nouvelles sur
l'analyse infinitesimale_ (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, _Traite d'analyse_, t. ii. (Paris,
1893). For elliptic functions there is an introductory book, P. Appell
and E. Lacour, _Principes de la theorie des fonctions elliptiques et
applications_ (Paris, 1897), beside the treatises of G.H. Halphen,
_Traite des fonctions elliptiques et de leurs applications_ (three
parts, Paris, 1886 ff.), and J. Tannery et J. Molk, _Elements de la
theorie 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, _Uber 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, _Theorie des fonctions
algebriques et de leurs integrales_; 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._ (Konigsberg, 1828); for the general
theory of theta functions, consult W. Wirtinger, _Untersuchungen uber
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 alterer 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, _Theorie
des fonctions algebriques de deux variables independantes_ (Paris,
1897-1906). This work contains, as Note v. t. ii. p. 485, a valuable
summary by MM. Castelnuovo and Enriques, _Sur quelques resultats
nouveaux dans la theorie des surfaces algebriques_, 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.)

FOOTNOTE:

[1] The word "function" (from Lat. _fungi_, to perform) has many
uses, with the fundamental sense of an activity special or proper to
an office, business or profession, or to an organ of an animal or
plant, the definite work for which the organ is an apparatus. From
the use of the word, as in the Italian _funzione_, for a ceremony of
the Roman Church, "function" is often employed for a public ceremony
of any kind, and loosely of a social entertainment or gathering.

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 Francaise.

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.