Chapter XV: Part 15

We may examine in this way the behaviour of the integrals at all the
points at which any one of the rational functions a1 ... an becomes
infinite; in general we must expect that beside these the value x =
[oo] will be a singular point for the solutions of the differential
equation. To test this we put x = 1/t throughout, and examine as
before at t = 0. For instance, the ordinary linear equation with
constant coefficients has no singular point for finite values of x; at
x = [oo] it has a singular point and is not regular; or again,
Bessel's equation x^2 + xy' + (x^2 - n^2)y = 0 is regular about x = 0,
but not about x = [oo]. An equation regular at all the finite
singularities and also at x = [oo] is called a Fuchsian equation. We
proceed to examine particularly the case of an equation of the second
order

y" + ay' + by = 0.

Putting x = 1/t, it becomes

d^2y/dt^2 + (2t^-1 - at^-2)dy/dt + bt^-4 y = 0,

which is not regular about t = 0 unless 2 - at^-1 and bt^-2, that is,
unless ax and bx^2 are finite at x =[oo]; which we thus assume; putting
y = t^r(1 + A1t + ... ), we find for the index equation at x =
[inifinity] the equation r(r - 1) + r(2 - ax)_0 + (bx^2)_0 = 0. If
there be finite singular points at [xi]1, ... [xi]m, where we assume
m>1, the cases m = 0, m = 1 being easily dealt with, and if [phi](x) =
(x - [xi]1) ... (x -[xi]m), we must have a.[phi](x) and b.[[phi](x)]^2
finite for all finite values of x, equal say to the respective
polynomials [psi](x) and [theta](x), of which by the conditions at x =
[oo] the highest respective orders possible are m - 1 and 2(m - 1).
The index equation at x = [xi]1 is r(r - 1) +
r[psi]([xi]1)/[phi]'([xi]1) + [theta]([xi])1/[[phi]'([xi]1)]^2 = 0, and
if [alpha]1, [beta]1 be its roots, we have [alpha]1 + [beta]1 = 1 -
[psi]([xi]1)/[phi]'([xi]1) and [alpha]1[beta]1 =
[theta]([xi])1/[[phi]'([xi]1)]^2. Thus by an elementary theorem of
algebra, the sum [Sigma](1 - [alpha]i - [beta]i)/(x - [xi]i), extended
to the m finite singular points, is equal to [psi](x)/[phi](x), and
the sum [Sigma](1 - [alpha]i - [beta]i) is equal to the ratio of the
coefficients of the highest powers of x in [psi](x) and [phi](x), and
therefore equal to 1 + [alpha] + [beta], where [alpha], [beta] are the
indices at x = [oo]. Further, if (x, 1)m-2 denote the integral part of
the quotient [theta](x)/[phi](x), we have
[Sigma][alpha]_i[beta]_i[phi]'([xi]_i)/(x - [xi]_i) equal to -(x,
1)_m-2 + [theta](x)/[phi](x), and the coefficient of x^m-2 in (x,
1)_m-2 is [alpha][beta]. Thus the differential equation has the form

y" + y'[Sigma](1 - [alpha]_i - [beta]_i)/(x - [xi]_i) + y[(x, 1)_m-2 +
[Sigma][alpha]_i[beta]_i[phi]'([xi]_i)/(x - [xi]_i)]/[phi](x) = 0.

If, however, we make a change in the dependent variable, putting y =
(x - [xi]1)^[alpha]1 ... (x - [xi]_m)^[alpha] m[eta], it is easy to
see that the equation changes into one having the same singular points
about each of which it is regular, and that the indices at x = [xi]_i
become 0 and [beta]_i - [alpha]_i, which we shall denote by [lambda]i,
for (x -[xi]_i)^[alpha]j can be developed in positive integral powers
of x -[xi]_i about x = [xi]_i; by this transformation the indices at x
= [oo] are changed to

[alpha] + [alpha]1 + ... + [alpha]m, [beta] + [beta]1 + ... + [beta]m

which we shall denote by [lambda], [mu]. If we suppose this change to
have been introduced, and still denote the independent variable by y,
the equation has the form

y" + y'[Sigma](1 - [lambda]_i)/(x - [xi]_i) + y(x, 1)_m-2/[phi](x) = 0,

while [lambda] + [mu] + [lambda]1 + ... + [lambda]_m = m - 1.
Conversely, it is easy to verify that if [lambda][mu] be the
coefficient of x^m-2 in (x, 1)_m-2, this equation has the specified
singular points and indices whatever be the other coefficients in (x,
1)_m-2.

Hypergeometric equation.

Thus we see that (beside the cases m = 0, m = 1) the "Fuchsian
equation" of the second order with _two_ finite singular points is
distinguished by the fact that it has a definite form when the
singular points and the indices are assigned. In that case, putting (x
- [xi]1)/(x - [xi]2) = t/(t - 1), the singular points are transformed
to 0, 1, [oo], and, as is clear, without change of indices. Still
denoting the independent variable by x, the equation then has the form

x(1 - x)y" + y'[1 - [lambda]1 - x(1 + [lambda] + [mu])] - [lambda][mu]y = 0,

which is the ordinary hypergeometric equation. Provided none of
[lambda]1, [lambda]2, [lambda] - [mu] be zero or integral about x = 0,
it has the solutions

F([lambda], [mu], 1 - [lambda]1, x), x^[lambda]1 F([lambda] + [lambda]1, [mu] + [lambda]1, 1 + [lambda]1, x);

about x = 1 it has the solutions

F([lambda], [mu], 1 - [lambda]2, 1 - x), (1 - x)^[lambda]1 F([lambda] + [lambda]2, [mu] + [lambda]2, 1 + [lambda]2, 1 - x),

where [lambda] + [mu] + [lambda]1 + [lambda]2 = 1; about x = [oo] it
has the solutions

x^-[lambda] F([lambda], [lambda] + [lambda]1, [lambda] - [mu] + 1, x^-1),
x^-[mu] F([mu], [mu] + [lambda]1, [mu] - [lambda] + 1, x^-1),

where F([alpha], [beta], [gamma], x) is the series

[alpha][beta]x [alpha]([alpha] + 1)[beta]([beta] + 1)x^2
1 + -------------- + ----------------------------------------- ...,
[gamma] 1.2.[gamma]([gamma] + 1)

which converges when |x| < 1, whatever [alpha], [beta], [gamma] may
be, converges for all values of x for which |x| = 1 provided the real
part of [gamma] - [alpha] - [beta] < 0 algebraically, and converges
for all these values except x = 1 provided the real part of [gamma] -
[alpha] -[beta] > -1 algebraically.

In accordance with our general theory, logarithms are to be expected
in the solution when one of [lambda]1, [lambda]2, [lambda] - [mu] is
zero or integral. Indeed when [lambda]1 is a negative integer, not
zero, the second solution about x = 0 would contain vanishing factors
in the denominators of its coefficients; in case [lambda] or [mu] be
one of the positive integers 1, 2, ... (-[lambda]1), vanishing factors
occur also in the numerators; and then, in fact, the second solution
about x = 0 becomes x^[lambda]1 times an integral polynomial of degree
(-[lambda]1) - [lambda] or of degree (-[lambda]1) - [mu]. But when
[lambda]1 is a negative integer including zero, and neither [lambda]
nor [mu] is one of the positive integers 1, 2 ... (-[lambda]1), the
second solution about x = 0 involves a term having the factor log x.
When [lambda]1 is a positive integer, not zero, the second solution
about x = 0 persists as a solution, in accordance with the order of
arrangement of the roots of the index equation in our theory; the
first solution is then replaced by an integral polynomial of degree
-[lambda] or -[mu]1, when [lambda] or [mu] is one of the negative
integers 0, -1, -2, ..., 1 - [lambda]1, but otherwise contains a
logarithm. Similarly for the solutions about x = 1 or x = [oo]; it
will be seen below how the results are deducible from those for x = 0.

March of the Integral.

Denote now the solutions about x = 0 by u1, u2; those about x = 1 by
v1, v2; and those about x = [oo] by w1, w2; in the region (S0S1)
common to the circles S0, S1 of radius 1 whose centres are the points
x = 0, x = 1, all the first four are valid, and there exist equations
u1 =Av1 + Bv2, u2 = Cv1 + Dv2 where A, B, C, D are constants; in the
region (S1S) lying inside the circle S1 and outside the circle S0,
those that are valid are v1, v2, w1, w2, and there exist equations v1
= Pw1 + Qw2, v2 = Rw1 + Tw2, where P, Q, R, T are constants; thus
considering any integral whose expression within the circle S0 is au1
+ bu2, where a, b are constants, the same integral will be represented
within the circle S1 by (aA + bC)v1 + (aB + bD)v2, and outside these
circles will be represented by

[(aA + bC)P + (aB + bD)R]w1 + [(aA + bC)Q + (aB + bD)T]w2.

A single-valued branch of such integral can be obtained by making a
barrier in the plane joining [oo] to 0 and 1 to [oo]; for instance, by
excluding the consideration of real negative values of x and of real
positive values greater than 1, and defining the phase of x and x - 1
for real values between 0 and 1 as respectively 0 and [pi].

Transformation of the equation into itself.

We can form the Fuchsian equation of the second order with three
arbitrary singular points [xi]1, [xi]2, [xi]3, and no singular point
at x = [oo], and with respective indices [alpha]1, [beta]1, [alpha]2,
[beta]2, [alpha]3, [beta]3 such that [alpha]1 + [beta]1 + [alpha]2 +
[beta]2 + [alpha]3 + [beta]3 = 1. This equation can then be
transformed into the hypergeometric equation in 24 ways; for out of
[xi]1, [xi]2, [xi]3 we can in six ways choose two, say [xi]1, [xi]2,
which are to be transformed respectively into 0 and 1, by (x -
[xi]1)/(x - [xi]2) = t(t - 1); and then there are four possible
transformations of the dependent variable which will reduce one of the
indices at t = 0 to zero and one of the indices at t = 1 also to zero,
namely, we may reduce either [alpha]1 or [beta]1 at t = 0, and
simultaneously either [alpha]2 or [beta]2 at t = 1. Thus the
hypergeometric equation itself can be transformed into itself in 24
ways, and from the expression F([lambda], [mu], 1 - [lambda]1, x)
which satisfies it follow 23 other forms of solution; they involve
four series in each of the arguments, x, x-1, 1/x, 1/(1-x), (x-1)/x,
x/(x-1). Five of the 23 solutions agree with the fundamental solutions
already described about x = 0, x = 1, x = [oo]; and from the
principles by which these were obtained it is immediately clear that
the 24 forms are, in value, equal in fours.

Inversion. Modular functions.

The quarter periods K, K' of Jacobi's theory of elliptic functions, of
which K = [int] [0 to [pi]/2] (1 - h sin^2[theta])^-1/2 d[theta], and K'
is the same function of 1-h, can easily be proved to be the solutions
of a hypergeometric equation of which h is the independent variable.
When K, K' are regarded as defined in terms of h by the differential
equation, the ratio K'/K is an infinitely many valued function of h.
But it is remarkable that Jacobi's own theory of theta functions leads
to an expression for h in terms of K'/K (see FUNCTION) in terms of
single-valued functions. We may then attempt to investigate, in
general, in what cases the independent variable x of a hypergeometric
equation is a single-valued function of the ratio s of two independent
integrals of the equation. The same inquiry is suggested by the
problem of ascertaining in what cases the hypergeometric series
F([alpha], [beta], [gamma], x) is the expansion of an algebraic
(irrational) function of x. In order to explain the meaning of the
question, suppose that the plane of x is divided along the real axis
from -[oo] to 0 and from 1 to +[oo], and, supposing logarithms not to
enter about x = 0, choose two quite definite integrals y1, y2 of the
equation, say

y1 = F([lambda], [mu], 1-[lambda]1, x),
y2 = x^[lambda]1 F([lambda] + [lambda]1, [mu] + [lambda]1, 1 + [lambda]1, x),

with the condition that the phase of x is zero when x is real and
between 0 and 1. Then the value of [sigma] = y2/y1 is definite for all
values of x in the divided plane, [sigma] being a single-valued
monogenic branch of an analytical function existing and without
singularities all over this region. If, now, the values of [sigma]
that so arise be plotted on to another plane, a value p + iq of
[sigma] being represented by a point (p, q) of this [stigma]-plane, and
the value of x from which it arose being mentally associated with this
point of the [sigma]-plane, these points will fill a connected region
therein, with a continuous boundary formed of four portions
corresponding to the two sides of the two barriers of the x-plane. The
question is then, firstly, whether the same value of s can arise for
two different values of x, that is, whether the same point (p, q) of
the [sigma]-plane can arise twice, or in other words, whether the
region of the [sigma]-plane overlaps itself or not. Supposing this is
not so, a second part of the question presents itself. If in the
x-plane the barrier joining -[oo] to 0 be momentarily removed, and x
describe a small circle with centre at x = 0 starting from a point x =
-h - ik, where h, k are small, real, and positive and coming back to
this point, the original value s at this point will be changed to a
value [sigma], which in the original case did not arise for this value
of x, and possibly not at all. If, now, after restoring the barrier
the values arising by continuation from [sigma] be similarly plotted
on the s-plane, we shall again obtain a region which, while not
overlapping itself, may quite possibly overlap the former region. In
that case two values of x would arise for the same value or values of
the quotient y2/y1, arising from two different branches of this
quotient. We shall understand then, by the condition that x is to be a
single-valued function of x, that the region in the [stimga]-plane
corresponding to any branch is not to overlap itself, and that no two
of the regions corresponding to the different branches are to overlap.
Now in describing the circle about x = 0 from x = -h - ik to -h + ik,
where h is small and k evanescent,

[stigma] = x^[lambda]1 F([lambda] + [lambda]1, [mu] + [lambda]1, 1 + [lambda]1, x)/F([lambda], [mu], 1 - [lambda]1, x)

is changed to [sigma] = [stigma]e^(2[pi]i[lambda])1. Thus the two
portions of boundary of the s-region corresponding to the two sides of
the barrier (-[oo], 0) meet (at [sigmaf] = 0 if the real part of
[lambda]1 be positive) at an angle 2[pi]L1, where L1 is the absolute
value of the real part of [lambda]1; the same is true for the
[sigma]-region representing the branch [sigma]. The condition that the
s-region shall not overlap itself requires, then, L1 = 1. But,
further, we may form an infinite number of branches [sigma] =
[stigma]e^(2[pi]i[lambda])1, [sigma]1 = e^(2[pi]i[lambda])1, ... in
the same way, and the corresponding regions in the plane upon which
y2/y1 is represented will have a common point and each have an angle
2[pi]L1; if neither overlaps the preceding, it will happen, if L1 is
not zero, that at length one is reached overlapping the first, unless
for some positive integer [alpha] we have 2[pi][alpha]L1 = 2[pi], in
other words L1 = 1/a. If this be so, the branch [sigma]_a-1 =
[stigma]e^(2[pi]ia[lambda])1 will be represented by a region having
the angle at the common point common with the region for the branch
[stigma]; but not altogether coinciding with this last region unless
[lambda]1 be real, and therefore = [+-]1/a; then there is only a finite
number, a, of branches obtainable in this way by crossing the barrier
(-[oo], 0). In precisely the same way, if we had begun by taking the
quotient

[stigma]' = (x - 1)^[lambda]2 F([lambda] + [lambda]2, [mu] + [lambda]2, 1 + [lambda]2, 1 - x)/F([lambda], [mu], 1 - [lambda]2, 1 - x)

of the two solutions about x = 1, we should have found that x is not a
single-valued function of [stigma]' unless [lambda]2 is the inverse of
an integer, or is zero; as [stigma]' is of the form (A[stigma] +
B)/(C[stigma] + D), A, B, C, D constants, the same is true in our
case; equally, by considering the integrals about x = [oo] we find, as
a third condition necessary in order that x may be a single-valued
function of [stigma], that [lambda] - [mu] must be the inverse of an
integer or be zero. These three differences of the indices, namely,
[lambda]1, [lambda]2, [lambda] - [mu], are the quantities which enter
in the differential equation satisfied by x as a function of [stigma],
which is easily found to be

x111 3^2x^211
- ---- + -------- = 1/2(h - h1 - h2)x^-1 (x - 1)^-1 + 1/2h1 x^-2 + 1/2h2(x - 1)^-2,
x1^3 2x1^4

where x1 = dx/d[stigma], &c.; and h1 = 1 - y1^2, h2 = 1 - [lambda]2^2,
h3 = 1 - ([lambda] - [mu])^2. Into the converse question whether the
three conditions are sufficient to ensure (1) that the [stigma] region
corresponding to any branch does not overlap itself, (2) that no two
such regions overlap, we have no space to enter. The second question
clearly requires the inquiry whether the group (that is, the monodromy
group) of the differential equation is properly discontinuous. (See
GROUPS, THEORY OF.)

The foregoing account will give an idea of the nature of the function theories of differential equations; it appears essential not to exclude some explanation of a theory intimately related both to such theories and to transformation theories, which is a generalization of Galois's theory of algebraic equations. We deal only with the application to homogeneous linear differential equations.

Rationality group of a linear equation.

Irreducibility of a rational equation.

In general a function of variables x1, x2 ... is said to be rational
when it can be formed from them and the integers 1, 2, 3, ... by a
finite number of additions, subtractions, multiplications and
divisions. We generalize this definition. Assume that we have assigned
a fundamental series of quantities and functions of x, in which x
itself is included, such that all quantities formed by a finite number
of additions, subtractions, multiplications, divisions _and
differentiations in regard to x_, of the terms of this series, are
themselves members of this series. Then the quantities of this series,
and only these, are called _rational_. By a rational function of
quantities p, q, r, ... is meant a function formed from them and any
of the fundamental rational quantities by a finite number of the five
fundamental operations. Thus it is a function which would be called,
simply, rational if the fundamental series were widened by the
addition to it of the quantities p, q, r, ... and those derivable from
them by the five fundamental operations. A rational ordinary
differential equation, with x as independent and y as dependent
variable, is then one which equates to zero a rational function of y,
the order k of the differential equation being that of the highest
differential coefficient y^(k) which enters; only such equations are
here discussed. Such an equation P = 0 is called _irreducible_ when,
firstly, being arranged as an integral polynomial in y^(k), this
polynomial is not the product of other polynomials in y^(k) also of
rational form; and, secondly, the equation has no solution satisfying
also a rational equation of lower order. From this it follows that if
an irreducible equation P = 0 have one solution satisfying another
rational equation Q = 0 of the same or higher order, then all the
solutions of P = 0 also satisfy Q = 0. For from the equation P = 0 we
can by differentiation express y^(k+1), y^(k+2), ... in terms of x, y,
y^(1), ... , y^(k), and so put the function Q rationally in terms of
these quantities only. It is sufficient, then, to prove the result
when the equation Q = 0 is of the same order as P = 0. Let both the
equations be arranged as integral polynomials in y^(k); their
algebraic eliminant in regard to y^(k) must then vanish identically,
for they are known to have one common solution not satisfying an
equation of lower order; thus the equation P = 0 involves Q = 0 for
all solutions of P = 0.

The variant function for a linear equation.

Now let y^(n) = [alpha]1y^(n-1) + ... + [alpha]_n y be a given
rational homogeneous linear differential equation; let y1, ... yn be n
particular functions of x, unconnected by any equation with constant
coefficients of the form c1y1 + ... + cnyn = 0, all satisfying the
differential equation; let [eta]1, ... [eta]n be linear functions of
y1, ... yn, say [eta]i = A_i1 y1 + ... + A_in yn, where the constant
coefficients Aij have a non-vanishing determinant; write ([eta]) =
A(y), these being the equations of a general linear homogeneous group
whose transformations may be denoted by A, B, .... We desire to form a
rational function [phi]([eta]), or say [phi](A(y)), of [eta]1, ...
[eta], in which the [eta]^2 constants Aij shall all be essential, and
not reduce effectively to a fewer number, as they would, for instance,
if the y1, ... yn were connected by a linear equation with constant
coefficients. Such a function is in fact given, if the solutions y1,
... yn be developable in positive integral powers about x = a, by
[phi]([eta]) = [eta]1 + (x - a)^n[eta]2 + ... + (x - a)^(n-1)n[eta]n.
Such a function, V, we call a _variant_.

The resolvent eqution.

Then differentiating V in regard to x, and replacing [eta]i^(n) by
its value a1[eta]^(n-1) + ... + an[eta], we can arrange dV/dx, and
similarly each of d^2/dx^2 ... d^NV/dx^N, where N = n^2, as a linear
function of the N quantities [eta]1, ... [eta]n, ... [eta]1^(n-1), ...
[eta]n^(n-1), and thence by elimination obtain a linear differential
equation for V of order N with rational coefficients. This we denote
by F = 0. Further, each of [eta]1 ... [eta]n is expressible as a
linear function of V, dV/dx, ... d^(N-1)V/dx^(N-1), with rational
coefficients not involving any of the n^2 coefficients A_ij, since
otherwise V would satisfy a linear equation of order less than N,
which is impossible, as it involves (linearly) the n^2 arbitrary
coefficients Aij, which would not enter into the coefficients of the
supposed equation. In particular, y1 ,.. yn are expressible rationally
as linear functions of [omega], d[omega]/dx, ...
d^(N-1)[omega]/dx^(N-1), where [omega] is the particular function
[phi](y). Any solution W of the equation F = 0 is derivable from
functions [zeta]1, ... [zeta]n, which are linear functions of y1, ...
yn, just as V was derived from [eta]1, ... [eta]n; but it does not
follow that these functions [zeta]i, ... [zeta]n are obtained from y1,
... yn by a transformation of the linear group A, B, ... ; for it may
happen that the determinant d([zeta]1, ... [zeta]n)/(dy1, ... yn) is
zero. In that case [zeta]1, ... [zeta]n may be called a singular set,
and W a singular solution; it satisfies an equation of lower than the
N-th order. But every solution V, W, ordinary or singular, of the
equation F = 0, is expressible rationally in terms of [omega],
d[omega]/dx, ... d^(N-1)[omega]/dx^(N-1); we shall write, simply, V =
r([omega]). Consider now the rational irreducible equation of lowest
order, not necessarily a linear equation, which is satisfied by
[omega]; as y1, ... yn are particular functions, it may quite well be
of order less than N; we call it the _resolvent equation_, suppose it
of order p, and denote it by [gamma](v). Upon it the whole theory
turns. In the first place, as [gamma](v) = 0 is satisfied by the
solution [omega] of F = 0, all the solutions of [gamma](v) are
solutions F = 0, and are therefore rationally expressible by [omega];
any one may then be denoted by r([omega]). If this solution of F = 0
be not singular, it corresponds to a transformation A of the linear
group (A, B, ...), effected upon y1, ... yn. The coefficients Aij of
this transformation follow from the expressions before mentioned for
[eta]1 ... [eta]n in terms of V, dV/dx, d^2V/dx^2, ... by substituting V
= r([omega]); thus they depend on the p arbitrary parameters which
enter into the general expression for the integral of the equation
[gamma](v) = 0. Without going into further details, it is then clear
enough that the resolvent equation, being irreducible and such that
any solution is expressible rationally, with p parameters, in terms of
the solution [omega], enables us to define a linear homogeneous group
of transformations of y1 ... yn depending on p parameters; and every
operation of this (continuous) group corresponds to a rational
transformation of the solution of the resolvent equation. This is the
group called the _rationality group_, or the _group of
transformations_ of the original homogeneous linear differential
equation.

The group must not be confounded with a subgroup of itself, the
_monodromy group_ of the equation, often called simply the group of
the equation, which is a set of transformations, not depending on
arbitrary variable parameters, arising for one particular fundamental
set of solutions of the linear equation (see GROUPS, THEORY OF).

The fundamental theorem in regard to the rationality group.

The importance of the rationality group consists in three
propositions. (1) Any rational function of y1, ... yn which is
unaltered in value by the transformations of the group can be written
in rational form. (2) If any rational function be changed in form,
becoming a rational function of y1, ... yn, a transformation of the
group applied to its new form will leave its value unaltered. (3) Any
homogeneous linear transformation leaving unaltered the value of every
rational function of y1, ... yn which has a rational value, belongs to
the group. It follows from these that any group of linear homogeneous
transformations having the properties (1) (2) is identical with the
group in question. It is clear that with these properties the group
must be of the greatest importance in attempting to discover what
functions of x must be regarded as rational in order that the values
of y1 ... yn may be expressed. And this is the problem of solving the
equation from another point of view.

LITERATURE.--([alpha]) _Formal or Transformation Theories for
Equations of the First Order_:--E. Goursat, _Lecons sur l'integration
des equations aux derivees partielles du premier ordre_ (Paris, 1891);
E. v. Weber, _Vorlesungen uber das Pfaff'sche Problem und die Theorie
der partiellen Differentialgleichungen erster Ordnung_ (Leipzig,
1900); S. Lie und G. Scheffers, _Geometrie der
Beruhrungstransformationen_, Bd. i. (Leipzig, 1896); Forsyth, _Theory
of Differential Equations, Part i., Exact Equations and Pfaff's
Problem_ (Cambridge, 1890); S. Lie, "Allgemeine Untersuchungen uber
Differentialgleichungen, die eine continuirliche endliche Gruppe
gestatten" (Memoir), _Mathem. Annal._xxv. (1885), pp. 71-151; S. Lie
und G. Scheffers, _Vorlesungen uber Differentialgleichungen mit
bekannten infinitesimalen Transformationen_ (Leipzig, 1891). A very
full bibliography is given in the book of E. v. Weber referred to;
those here named are perhaps sufficiently representative of modern
works. Of classical works may be named: Jacobi, _Vorlesungen uber
Dynamik_ (von A. Clebsch, Berlin, 1866); _Werke, Supplementband_; G
Monge, _Application de l'analyse a la geometrie_ (par M. Liouville,
Paris, 1850); J. L. Lagrange, _Lecons sur le calcul des fonctions_
(Paris, 1806), and _Theorie des fonctions analytiques_ (Paris,
Prairial, an V); G. Boole, _A Treatise on Differential Equations_
(London, 1859); and _Supplementary Volume_ (London, 1865); Darboux,
_Lecons sur la theorie generale des surfaces_, tt. i.-iv. (Paris,
1887-1896); S. Lie, _Theorie der transformationsgruppen_ ii. (on
Contact Transformations) (Leipzig, 1890).

([beta]) _Quantitative or Function Theories for Linear Equations_:--C.
Jordan, _Cours d'analyse_, t. iii. (Paris, 1896); E. Picard, _Traite
d'analyse_, tt. ii. and iii. (Paris, 1893, 1896); Fuchs, _Various
Memoirs, beginning with that in Crelle's Journal_, Bd. lxvi. p. 121;
Riemann, _Werke_, 2^r Aufl. (1892); Schlesinger, _Handbuch der Theorie
der linearen Differentialgleichungen_, Bde. i.-ii. (Leipzig,
1895-1898); Heffter, _Einleitung in die Theorie der linearen
Differentialgleichungen mit einer unabhangigen Variablen_ (Leipzig,
1894); Klein, _Vorlesungen uber lineare Differentialgleichungen der
zweiten Ordnung_ (Autographed, Gottingen, 1894); and _Vorlesungen uber
die hypergeometrische Function_ (Autographed, Gottingen, 1894);
Forsyth, _Theory of Differential Equations, Linear Equations_.

([gamma]) _Rationality Group (of Linear Differential
Equations)_:--Picard, _Traite d'Analyse_, as above, t. iii.; Vessiot,
_Annales de l'Ecole Normale_, serie III. t. ix. p. 199 (Memoir); S.
Lie, _Transformationsgruppen_, as above, iii. A connected account is
given in Schlesinger, as above, Bd. ii., erstes Theil.

([delta]) _Function Theories of Non-Linear Ordinary
Equations_:--Painleve, _Lecons sur la theorie analytique des equations
differentielles_ (Paris, 1897, Autographed); Forsyth, _Theory of
Differential Equations, Part ii., Ordinary Equations not Linear_ (two
volumes, ii. and iii.) (Cambridge, 1900); Konigsberger, _Lehrbuch der
Theorie der Differentialgleichungen_ (Leipzig, 1889); Painleve,
_Lecons sur l'integration des equations differentielles de la
mecanique et applications_ (Paris, 1895).

([epsilon]) _Formal Theories of Partial Equations of the Second and
Higher Orders_:--E. Goursat, _Lecons sur l'integration des equations
aux derivees partielles du second ordre_, tt. i. and ii. (Paris, 1896,
1898); Forsyth, _Treatise on Differential Equations_ (London, 1889);
and _Phil. Trans. Roy. Soc._ (A.), vol. cxci. (1898), pp. 1-86.

([zeta]) See also the six extensive articles in the second volume of
the German _Encyclopaedia of Mathematics_. (H. F. BA.)

DIFFLUGIA (L. Leclerc), a genus of lobose Rhizopoda, characterized by a shell formed of sand granules cemented together; these are swallowed by the animal, and during the process of bud-fission they pass to the surface of the daughter-bud and are cemented there. _Centropyxis_ (Steia) and _Lecqueureuxia_ (Schlumberg) differ only in minor points.

DIFFRACTION OF LIGHT.--1. When light proceeding from a small source falls upon an opaque object, a shadow is cast upon a screen situated behind the obstacle, and this shadow is found to be bordered by alternations of brightness and darkness, known as "diffraction bands." The phenomena thus presented were described by Grimaldi and by Newton. Subsequently T. Young showed that in their formation interference plays an important part, but the complete explanation was reserved for A. J. Fresnel. Later investigations by Fraunhofer, Airy and others have greatly widened the field, and under the head of "diffraction" are now usually treated all the effects dependent upon the limitation of a beam of light, as well as those which arise from irregularities of any kind at surfaces through which it is transmitted, or at which it is reflected.

2. _Shadows._--In the infancy of the undulatory theory the objection most frequently urged against it was the difficulty of explaining the very existence of shadows. Thanks to Fresnel and his followers, this department of optics is now precisely the one in which the theory has gained its greatest triumphs. The principle employed in these investigations is due to C. Huygens, and may be thus formulated. If round the origin of waves an ideal closed surface be drawn, the whole action of the waves in the region beyond may be regarded as due to the motion continually propagated across the various elements of this surface. The wave motion due to any element of the surface is called a _secondary_ wave, and in estimating the total effect regard must be paid to the phases as well as the amplitudes of the components. It is usually convenient to choose as the surface of resolution a _wave-front_, i.e. a surface at which the primary vibrations are in one phase. Any obscurity that may hang over Huygens's principle is due mainly to the indefiniteness of thought and expression which we must be content to put up with if we wish to avoid pledging ourselves as to the character of the vibrations. In the application to sound, where we know what we are dealing with, the matter is simple enough in principle, although mathematical difficulties would often stand in the way of the calculations we might wish to make. The ideal surface of resolution may be there regarded as a flexible lamina; and we know that, if by forces locally applied every element of the lamina be made to move normally to itself exactly as the air at that place does, the external aerial motion is fully determined. By the principle of superposition the whole effect may be found by integration of the partial effects due to each element of the surface, the other elements remaining at rest.

We will now consider in detail the important case in which uniform
plane waves are resolved at a surface coincident with a wave-front
(OQ). We imagine a wave-front divided into elementary rings or
zones--often named after Huygens, but better after Fresnel--by spheres
described round P (the point at which the aggregate effect is to be
estimated), the first sphere, touching the plane at O, with a radius
equal to PO, and the succeeding spheres with radii increasing at each
step by 1/2[lambda]. There are thus marked out a series of circles,
whose radii x are given by x^2 + r^2 = (r + 1/2n[lambda])^2, or x^2 =
n[lambda]r nearly; so that the rings are at first of nearly equal
area. Now the effect upon P of each element of the plane is
proportional to its area; but it depends also upon the distance from
P, and possibly upon the inclination of the secondary ray to the
direction of vibration and to the wave-front.

O x Q
---------------------------
| /
| /
| /
| /
| /
| /
| /
r| /
| /
| /
| /
| /
| /
| /
| /
| /
P|/

FIG. 1.

The latter question can only be treated in connexion with the
dynamical theory (see below, S 11); but under all ordinary
circumstances the result is independent of the precise answer that may
be given. All that it is necessary to assume is that the effects of
the successive zones gradually diminish, whether from the increasing
obliquity of the secondary ray or because (on account of the
limitation of the region of integration) the zones become at last more
and more incomplete. The component vibrations at P due to the
successive zones are thus nearly equal in amplitude and opposite in
phase (the phase of each corresponding to that of the infinitesimal
circle midway between the boundaries), and the series which we have to
sum is one in which the terms are alternately opposite in sign and,
while at first nearly constant in numerical magnitude, gradually
diminish to zero. In such a series each term may be regarded as very
nearly indeed destroyed by the halves of its immediate neighbours, and
thus the sum of the whole series is represented by half the first
term, which stands over uncompensated. The question is thus reduced to
that of finding the effect of the first zone, or central circle, of
which the area is [pi][lambda]r.

We have seen that the problem before us is independent of the law of
the secondary wave as regards obliquity; but the result of the
integration necessarily involves the law of the intensity and phase of
a secondary wave as a function of r, the distance from the origin. And
we may in fact, as was done by A. Smith (_Camb. Math. Journ._, 1843,
3, p. 46), determine the law of the secondary wave, by comparing the
result of the integration with that obtained by supposing the primary
wave to pass on to P without resolution.

Now as to the phase of the secondary wave, it might appear natural to
suppose that it starts from any point Q with the phase of the primary
wave, so that on arrival at P, it is retarded by the amount
corresponding to QP. But a little consideration will prove that in
that case the series of secondary waves could not reconstitute the
primary wave. For the aggregate effect of the secondary waves is the
half of that of the first Fresnel zone, and it is the central element
only of that zone for which the distance to be travelled is equal to
r. Let us conceive the zone in question to be divided into
infinitesimal rings of equal area. The effects due to each of these
rings are equal in amplitude and of phase ranging uniformly over half
a complete period. The phase of the resultant is midway between those
of the extreme elements, that is to say, a quarter of a period behind
that due to the element at the centre of the circle. It is accordingly
necessary to suppose that the secondary waves start with a phase
one-quarter of a period in advance of that of the primary wave at the
surface of resolution.

Further, it is evident that account must be taken of the variation of
phase in estimating the magnitude of the effect at P of the first
zone. The middle element alone contributes without deduction; the
effect of every other must be found by introduction of a resolving
factor, equal to cos [theta], if [theta] represent the difference of
phase between this element and the resultant. Accordingly, the
amplitude of the resultant will be less than if all its components had
the same phase, in the ratio

_ +1/2[pi]
/
| cos [theta]d[theta] : [pi],
_/-1/2[pi]

or 2 : [pi]. Now 2 area /[pi] = 2[lambda]r; so that, in order to
reconcile the amplitude of the primary wave (taken as unity) with the
half effect of the first zone, the amplitude, at distance r, of the
secondary wave emitted from the element of area dS must be taken to be

dS/[lambda]r (1).

By this expression, in conjunction with the quarter-period
acceleration of phase, the law of the secondary wave is determined.

That the amplitude of the secondary wave should vary as r^-1 was to be
expected from considerations respecting energy; but the occurrence of
the factor [lambda]^-1, and the acceleration of phase, have sometimes
been regarded as mysterious. It may be well therefore to remember that
precisely these laws apply to a secondary wave of sound, which can be
investigated upon the strictest mechanical principles.

The recomposition of the secondary waves may also be treated
analytically. If the primary wave at O be cos kat, the effect of the
secondary wave proceeding from the element dS at Q is

dS dS
------------- cos k(at - [rho] + 1/4[lambda]) = ------------- sin k(at - [rho]).
[lambda][rho] [lambda][rho]

If dS = 2[pi]xdx, we have for the whole effect

_[oo]
2[pi] / sin k(at - [rho])x dx
- -------- | ---------------------,
[lambda] _/ 0 [rho]

or, since xdx = [rho]d[rho], k = 2[pi]/[lambda],

_[oo] _ _
/ | |[oo]
-k | sin k(at - [rho])d[rho] = | -cos k(at - [rho])| .
_/r |_ _|r

In order to obtain the effect of the primary wave, as retarded by
traversing the distance r, viz. cos k(at - r), it is necessary to
suppose that the integrated term vanishes at the upper limit. And it
is important to notice that without some further understanding the
integral is really ambiguous. According to the assumed law of the
secondary wave, the result must actually depend upon the precise
radius of the outer boundary of the region of integration, supposed to
be exactly circular. This case is, however, at most very special and
exceptional. We may usually suppose that a large number of the outer
rings are incomplete, so that the integrated term at the upper limit
may properly be taken to vanish. If a formal proof be desired, it may
be obtained by introducing into the integral a factor such as
e^-h[rho], in which h is ultimately made to diminish without limit.

When the primary wave is plane, the area of the first Fresnel zone is
[pi][lambda]r, and, since the secondary waves vary as r^-1, the
intensity is independent of r, as of course it should be. If, however,
the primary wave be spherical, and of radius a at the wave-front of
resolution, then we know that at a distance r further on the amplitude
of the primary wave will be diminished in the ratio a:(r + a). This
may be regarded as a consequence of the altered area of the first
Fresnel zone. For, if x be its radius, we have

/
{(r + 1/2[lambda])^2 - x^2} + \/ {a^2 - x^2} = r + a,

so that

x^2 = [lambda]ar/(a + r) nearly.

Since the distance to be travelled by the secondary waves is still r,
we see how the effect of the first zone, and therefore of the whole
series is proportional to a/(a + r). In like manner may be treated
other cases, such as that of a primary wave-front of unequal principal
curvatures.

The general explanation of the formation of shadows may also be
conveniently based upon Fresnel's zones. If the point under
consideration be so far away from the geometrical shadow that a large
number of the earlier zones are complete, then the illumination,
determined sensibly by the first zone, is the same as if there were no
obstruction at all. If, on the other hand, the point be well immersed
in the geometrical shadow, the earlier zones are altogether missing,
and, instead of a series of terms beginning with finite numerical
magnitude and gradually diminishing to zero, we have now to deal with
one of which the terms diminish to zero _at both ends_. The sum of
such a series is very approximately zero, each term being neutralized
by the halves of its immediate neighbours, which are of the opposite
sign. The question of light or darkness then depends upon whether the
series begins or ends abruptly. With few exceptions, abruptness can
occur only in the presence of the first term, viz. when the secondary
wave of least retardation is unobstructed, or when a _ray_ passes
through the point under consideration. According to the undulatory
theory the light cannot be regarded strictly as travelling along a
ray; but the existence of an unobstructed ray implies that the system
of Fresnel's zones can be commenced, and, if a large number of these
zones are fully developed and do not terminate abruptly, the
illumination is unaffected by the neighbourhood of obstacles.
Intermediate cases in which a few zones only are formed belong
especially to the province of diffraction.

An interesting exception to the general rule that full brightness
requires the existence of the first zone occurs when the obstacle
assumes the form of a small circular disk parallel to the plane of the
incident waves. In the earlier half of the 18th century R. Delisle
found that the centre of the circular shadow was occupied by a bright
point of light, but the observation passed into oblivion until S. D.
Poisson brought forward as an objection to Fresnel's theory that it
required at the centre of a circular shadow a point as bright as if no
obstacle were intervening. If we conceive the primary wave to be
broken up at the plane of the disk, a system of Fresnel's zones can be
constructed which begin from the circumference; and the first zone
external to the disk plays the part ordinarily taken by the centre of
the entire system. The whole effect is the half of that of the first
existing zone, and this is sensibly the same as if there were no
obstruction.

When light passes through a small circular or annular aperture, the
illumination at any point along the axis depends upon the precise
relation between the aperture and the distance from it at which the
point is taken. If, as in the last paragraph, we imagine a system of
zones to be drawn commencing from the inner circular boundary of the
aperture, the question turns upon the manner in which the series
terminates at the outer boundary. If the aperture be such as to fit
exactly an integral number of zones, the aggregate effect may be
regarded as the half of those due to the first and last zones. If the
number of zones be even, the action of the first and last zones are
antagonistic, and there is complete darkness at the point. If on the
other hand the number of zones be odd, the effects conspire; and the
illumination (proportional to the square of the amplitude) is four
times as great as if there were no obstruction at all.

The process of augmenting the resultant illumination at a particular
point by stopping some of the secondary rays may be carried much
further (Soret, _Pogg. Ann._, 1875, 156, p. 99). By the aid of
photography it is easy to prepare a plate, transparent where the zones
of odd order fall, and opaque where those of even order fall. Such a
plate has the power of a condensing lens, and gives an illumination
out of all proportion to what could be obtained without it. An even
greater effect (fourfold) can be attained by providing that the
stoppage of the light from the alternate zones is replaced by a
phase-reversal without loss of amplitude. R. W. Wood (_Phil. Mag._,
1898, 45, p 513) has succeeded in constructing zone plates upon this
principle.

In such experiments the narrowness of the zones renders necessary a
pretty close approximation to the geometrical conditions. Thus in the
case of the circular disk, equidistant (r) from the source of light
and from the screen upon which the shadow is observed, the width of
the first exterior zone is given by

dx = [lambda](2r)/4(2x),

2x being the diameter of the disk. If 2r = 1000 cm., 2x = 1 cm.,
[lambda] = 6 X 10^-5 cm., then dx = .0015 cm. Hence, in order that
this zone may be perfectly formed, there should be no error in the
circumference of the order of .001 cm. (It is easy to see that the
radius of the bright spot is of the same order of magnitude.) The
experiment succeeds in a dark room of the length above mentioned, with
a threepenny bit (supported by three threads) as obstacle, the origin
of light being a small needle hole in a plate of tin, through which
the sun's rays shine horizontally after reflection from an external
mirror. In the absence of a heliostat it is more convenient to obtain
a point of light with the aid of a lens of short focus.

The amplitude of the light at any point in the axis, when plane waves
are incident perpendicularly upon an annular aperture, is, as above,

cos k(at - r1) - cos k(at - r2) = 2 sin kat sin k(r1 - r2),

r2, r1 being the distances of the outer and inner boundaries from the
point in question. It is scarcely necessary to remark that in all such
cases the calculation applies in the first instance to homogeneous
light, and that, in accordance with Fourier's theorem, each
homogeneous component of a mixture may be treated separately. When the
original light is white, the presence of some components and the
absence of others will usually give rise to coloured effects, variable
with the precise circumstances of the case.

Although the matter can be fully treated only upon the basis of a
dynamical theory, it is proper to point out at once that there is an
element of assumption in the application of Huygens's principle to the
calculation of the effects produced by opaque screens of limited
extent. Properly applied, the principle could not fail; but, as may
readily be proved in the case of sonorous waves, it is not in
strictness sufficient to assume the expression for a secondary wave
suitable when the primary wave is undisturbed, with mere limitation of
the integration to the transparent parts of the screen. But, except
perhaps in the case of very fine gratings, it is probable that the
error thus caused is insignificant; for the incorrect estimation of
the secondary waves will be limited to distances of a few wave-lengths
only from the boundary of opaque and transparent parts.

3. _Fraunhofer's Diffraction Phenomena._--A very general problem in diffraction is the investigation of the distribution of light over a screen upon which impinge divergent or convergent spherical waves after passage through various diffracting apertures. When the waves are convergent and the recipient screen is placed so as to contain the centre of convergency--the image of the original radiant point, the calculation assumes a less complicated form. This class of phenomena was investigated by J. von Fraunhofer (upon principles laid down by Fresnel), and are sometimes called after his name. We may conveniently commence with them on account of their simplicity and great importance in respect to the theory of optical instruments.

If f be the radius of the spherical wave at the place of resolution,
where the vibration is represented by cos kat, then at any point M
(fig. 2) in the recipient screen the vibration due to an element dS of
the wave-front is (S 2)

dS
- ------------- sin k(at - [rho]),
[lambda][rho]

[rho] being the distance between M and the element dS.

Taking co-ordinates in the plane of the screen with the centre of the
wave as origin, let us represent M by [xi], [eta], and P (where dS is
situated) by x, y, z. Then

[rho]^2 = (x - [xi])^2 + (y - [eta])^2 + z^2, f^2 = x^2 + y^2 + z^2;

so that

[rho]^2 = f^2 - 2x[xi] - 2y[eta] + [xi]^2 + [eta]^2.

In the applications with which we are concerned, [xi], [eta] are very
small quantities; and we may take

/ x[xi] + y[eta]\
[rho] = f ( 1 - -------------- ).
\ f^2 /

At the same time dS may be identified with dxdy, and in the
denominator [rho] may be treated as constant and equal to f. Thus the
expression for the vibration at M becomes

_ _
1 / / / x[xi] + y[eta]\
- --------------- | | sin k ( at - f + -------------- ) dxdy (1);
[lambda]^2[f]^2 _/_/ \ f /

and for the intensity, represented by the square of the amplitude,

_ _ _ _
1 | / / x[xi] + y[eta] |^2
I^2 = ------------- | | | sin k -------------- dxdy |
[lambda]^2f^2 |_ _/_/ f _|
_ _ _ _
1 | / / x[xi] + y[eta] |^2
+ ------------- | | | cos k -------------- dxdy | (2).
[lambda]^2f^2 |_ _/_/ f _|

This expression for the intensity becomes rigorously applicable when f
is indefinitely great, so that ordinary optical aberration disappears.
The incident waves are thus plane, and are limited to a plane aperture
coincident with a wave-front. The integrals are then properly
functions of the _direction_ in which the light is to be estimated.

In experiment under ordinary circumstances it makes no difference
whether the collecting lens is in front of or behind the diffracting
aperture. It is usually most convenient to employ a telescope focused
upon the radiant point, and to place the diffracting apertures
immediately in front of the object-glass. What is seen through the
eye-piece in any case is the same as would be depicted upon a screen
in the focal plane.

Before proceeding to special cases it may be well to call attention to
some general properties of the solution expressed by (2) (see Bridge,
_Phil. Mag._, 1858).

If when the aperture is given, the wave-length (proportional to k^-1)
varies, the composition of the integrals is unaltered, provided [xi]
and [eta] are taken universely proportional to [lambda]. A diminution
of [lambda] thus leads to a simple proportional shrinkage of the
diffraction pattern, attended by an augmentation of brilliancy in
proportion to [lambda]^-2.

If the wave-length remains unchanged, similar effects are produced by
an increase in the scale of the aperture. The linear dimension of the
diffraction pattern is inversely as that of the aperture, and the
brightness at corresponding points is as the _square_ of the area of
aperture.

If the aperture and wave-length increase in the same proportion, the
size and shape of the diffraction pattern undergo no change.

We will now apply the integrals (2) to the case of a rectangular
aperture of width a parallel to x and of width b parallel to y. The
limits of integration for x may thus be taken to be -1/2a and +1/2a,
and for y to be -1/2b, +1/2b. We readily find (with substitution for k
of 2[pi]/[lambda])

[pi]a[xi] [pi]b[eta]
sin^2 --------- sin^2 ----------
a^2b^2 f[lambda] f[lambda]
I^2 = ------------ . ----------------- . ----------------- (3),
f^2[lambda]^2 [pi]^2a^2[xi]^2 [pi]^2b^2[eta]^2
-------------- -------------
f^2[lambda]^2 f^2[lambda]^2

as representing the distribution of light in the image of a
mathematical point when the aperture is rectangular, as is often the
case in spectroscopes.

The second and third factors of (3) being each of the form sin^2u/u^2,
we have to examine the character of this function. It vanishes when u
= m[pi], m being any whole number other than zero. When u = 0, it
takes the value unity. The maxima occur when

u = tan u, (4),

and then

sin^2u/u^2 = cos^2u (5).

To calculate the roots of (5) we may assume

u = (m + 1/2)[pi] - y = U - y,

where y is a positive quantity which is small when u is large.
Substituting this, we find cot y = U - y, whence

1 / y y- \ y^3 2y^5 17y^7
y = - ( 1 + - + --- + ... ) - --- ---- - -----.
U \ U U^2 / 3 15 315

This equation is to be solved by successive approximation. It will
readily be found that

2 13 146
u = U - y = U - U^-1 - -- U^-3 - -- U^-5 - --- U^-7 - ... (6).
3 15 105