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
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