Chapter XIV: Part 14

It can at once be verified that for any three functions
[f[[phi][psi]]] + [[phi][psi]f]] + [[psi][f[phi]]] = df/dz
[[phi][psi]] + d[phi]/dz [[psi]f] + d[psi]/dz [f[phi]], which when f,
[phi],[psi] do not contain z becomes the identity (f([phi][psi])) +
(phi([psi]f)) + ([psi](f[phi])) = 0. Then, if X1, ... Xn, P1, ... Pn
be such functions Of x1, ... xn, p1 ... pn that P1 dX1 + ... + Pn dXn
is identically equal to p1dx1 + ... + pn dxn, it can be shown by
elementary algebra, after equating coefficients of independent
differentials, (1) that the functions X1, ... Pn are independent
functions of the 2n variables x1, ... pn, so that the equations x'i =
Xi, p'i = Pi can be solved for x1, ... xn, p1, ... pn, and represent
therefore a transformation, which we call a homogeneous contact
transformation; (2) that the X1, ... Xn are homogeneous functions of
p1, ... pn of zero dimensions, the P1, ... Pn are homogeneous
functions of p1, ... pn of dimension one, and the 1/2n(n - 1) relations
(Xi Xj) = 0 are verified. So also are the n^2 relations (Pi Xi) = 1,
(Pi Xj) = 0, (Pi Pj) = 0. Conversely, if X1, ... Xn be independent
functions, each homogeneous of zero dimension in p1, ... pn satisfying
the 1/2n(n - 1) relations (Xi Xj) = 0, then P1, ... Pn can be uniquely
determined, by solving linear algebraic equations, such that P1 dX1 +
... + Pn dXn = p1 dx1 + ... + pn dxn. If now we put n + 1 for n, put z
for x_n+1, Z for X_n+1, Qi for -Pi/P_n+1, for i = 1, ... n, put qi for
-p_i/p_n+1 and [sigma] for q_n+1/Q_n+1, and then finally write P1, ...
Pn, p1, ... pn for Q1, ... Qn, q1, ... qn, we obtain the following
results: If ZX1 ... Xn, P1, ... Pn be functions of z, x1, ... xn, p1,
... pn, such that the expression dZ - P1 dX1 - ... - Pn dXn is
identically equal to [sigma](dz - p1 dx1 - ... - pn dxn), and [sigma]
not zero, then (1) the functions Z, X1, ... Xn, P1, ... Pn are
independent functions of z, x1, ... xn, p1, ... pn, so that the
equations z' = Z, x'i = Xi, p'i = Pi can be solved for z, x1, ... xn,
p1, ... pn and determine a transformation which we call a
(non-homogeneous) contact transformation; (2) the Z, X1, ... Xn verify
the 1/2n(n + 1) identities [Z Xi] = 0, [Xi Xj] = 0. And the further
identities

[Pi Xi] = [sigma], [Pi Xj] = 0, [Pi Z] = [sigma]Pi, [Pi Pj] = 0,

dZ dXi dPi
[Z[sigma]] = [sigma]-- - [sigma]^2, [Xi [sigma]] = [sigma]---, [Pi [sigma]] = [sigma]---
dz dz dz

are also verified. Conversely, if Z, x1, ... Xn be independent
functions satisfying the identities [Z Xi] = 0, [Xi Xj] = 0, then
[sigma], other than zero, and P1, ... Pn can be uniquely determined,
by solution of algebraic equations, such that

dZ - P1 dX1 - ... - Pn dXn = [sigma](dz - p1 dx1 - ... - p_n dx_n).

Finally, there is a particular case of great importance arising when
[sigma] = 1, which gives the results: (1) If U, X1, ... Xn, P1, ... Pn
be 2n + 1 functions of the 2n independent variables x1, ... xn, p1,
... pn, satisfying the identity

dU + P1 dx1 + ... + Pn dXn = p1 dx1 + ... + p_n dx_n,

then the 2n functions P1, ... Pn, X1, ... Xn are independent, and we
have

(Xi Xj) = 0, (Xi U) = [delta]Xi, (Pi Xi) = 1, (Pi Xj) = 0, (Pi Pj ) = 0, (Pi U) + Pi = [delta]Pi,

where [delta] denotes the operator p1d/dp1 + ... + pnd/dpn; (2) If X1,
... Xn be independent functions of x1, ... xn, p1, ... pn, such that
(Xi Xj) = 0, then U can be found by a quadrature, such that

(Xi U) = [delta]Xi;

and when Xi, ... Xn, U satisfy these 1/2n(n + 1) conditions, then P1,
... Pn can be found, by solution of linear algebraic equations, to
render true the identity dU + P1 dX1 + ... + Pn dXn = p1 dx1 + ... +
pn dxn; (3) Functions X1, ... Xn, P1, ... Pn can be found to satisfy
this differential identity when U is an arbitrary given function of
x1, ... xn, p1, ... pn; but this requires integrations. In order to
see what integrations, it is only necessary to verify the statement
that if U be an arbitrary given function of x1, ... xn, p1, ... pn,
and, for r < n, X1, ... Xr be independent functions of these
variables, such that (X_[sigma] U) = [delta]X_[sigma], (X_[rho]
X_[sigma]) = 0, for [rho], [sigma] = 1 ... r, then the r + 1
homogeneous linear partial differential equations of the first order
(Uf) + [delta]f = 0, (X[rho]f) = 0, form a complete system. It will be
seen that the assumptions above made for the reduction of Pfaffian
expressions follow from the results here enunciated for contact
transformations.

Partial differential equation of the first order.

Meaning of a solution of the equation.

We pass on now to consider the solution of any partial differential equation of the first order; we attempt to explain certain ideas relatively to a single equation with any number of independent variables (in particular, an ordinary equation of the first order with one independent variable) by speaking of a single equation with two independent variables x, y, and one dependent variable z. It will be seen that we are naturally led to consider systems of such simultaneous equations, which we consider below. The central discovery of the transformation theory of the solution of an equation F(x, y, z, dz/dx, dz/dy) = 0 is that its solution can always be reduced to the solution of partial equations which are _linear_. For this, however, we must regard dz/dx, dz/dy, during the process of integration, not as the differential coefficients of a function z in regard to x and y, but as variables independent of x, y, z, the too great indefiniteness that might thus appear to be introduced being provided for in another way. We notice that if z = [psi](x, y) be a solution of the differential equation, then dz = dxd[psi]/dx + dyd[psi]/dy; thus if we denote the equation by F(x, y, z, p, q,) = 0, and prescribe the condition dz = pdx + qdy for every solution, any solution such as z = [psi](x, y) will necessarily be associated with the equations p = dz/dx, q = dz/dy, and z will satisfy the equation in its original form. We have previously seen (under _Pfaffian Expressions_) that if five variables x, y, z, p, q, otherwise independent, be subject to dz - pdx - qdy = 0, they must in fact be subject to at least three mutual relations. If we associate with a point (x, y, z) the plane

Z - z = p(X - x) + q(Y - y)

passing through it, where X, Y, Z are current co-ordinates, and call this association a surface-element; and if two consecutive elements of which the point(x + dx, y + dy, z + dz) of one lies on the plane of the other, for which, that is, the condition dz = pdx + qdy is satisfied, be said to be _connected,_ and an infinity of connected elements following one another continuously be called a _connectivity_, then our statement is that a connectivity consists of not more than [oo]^2 elements, the whole number of elements (x, y, z, p, q) that are possible being called [oo]^5. The solution of an equation F(x, y, z, dz/dx, dz/dy) = 0 is then to be understood to mean finding in all possible ways, from the [oo]^4 elements (x, y, z, p, q) which satisfy F(x, y, z, p, q) = 0 a set of [oo]^2 elements forming a connectivity; or, more analytically, finding in all possible ways two relations G = 0, H = 0 connecting x, y, z, p, q and independent of F = 0, so that the three relations together may involve

dz = pdx + qdy.

Such a set of three relations may, for example, be of the form z = [psi](x, y), p = d[psi]/dx, q = d[psi]/dy; but it may also, as another case, involve two relations z = [psi](y), x = [psi]1(y) connecting x, y, z, the third relation being

[psi]'(y) = p[psi]'1(y) + q,

the connectivity consisting in that case, geometrically, of a curve in space taken with [oo]^1 of its tangent planes; or, finally, a connectivity is constituted by a fixed point and all the planes passing through that point. This generalized view of the meaning of a solution of F = 0 is of advantage, moreover, in view of anomalies otherwise arising from special forms of the equation itself. For instance, we may include the case, sometimes arising when the equation to be solved is obtained by transformation from another equation, in which F does not contain either p or q. Then the equation has [oo]^2 solutions, each consisting of an arbitrary point of the surface F = 0 and all the [oo]^2 planes passing through this point; it also has [oo]^2 solutions, each consisting of a curve drawn on the surface F = 0 and all the tangent planes of this curve, the whole consisting of [oo]^2 elements; finally, it has also an isolated (or singular) solution consisting of the points of the surface, each associated with the tangent plane of the surface thereat, also [oo]^2 elements in all. Or again, a linear equation F = Pp + Qq - R = 0, wherein P, Q, R are functions of x, y, z only, has [oo]^2 solutions, each consisting of one of the curves defined by

dx/P = dy/Q = dz/R

taken with all the tangent planes of this curve; and the same equation has [oo]^2 solutions, each consisting of the points of a surface containing [oo]^1 of these curves and the tangent planes of this surface. And for the case of n variables there is similarly the possibility of n + 1 kinds of solution of an equation F(x1, ... xn, z, p1, ... pn) = 0; these can, however, by a simple contact transformation be reduced to one kind, in which there is only one relation z' = [psi](x'1, ... x'n) connecting the new variables x'1, ... x'n, z' (see under PFAFFIAN EXPRESSIONS); just as in the case of the solution

z = [psi](y), x = [psi]1(y), [psi]'(y) = p[psi]'1(y) + q

of the equation Pp + Qq = R the transformation z' = z - px, x' = p, p' = -x, y' = y, q' = q gives the solution

z' = [psi](y') + x'[psi]1(y'), p' = dz'/dx', q' = dz'/dy'

of the transformed equation. These explanations take no account of the possibility of p and q being infinite; this can be dealt with by writing p = -u/w, q = -v/w, and considering homogeneous equations in u, v, w, with udx + vdy + wdz = 0 as the differential relation necessary for a connectivity; in practice we use the ideas associated with such a procedure more often without the appropriate notation.

Order of the ideas.

In utilizing these general notions we shall first consider the theory of characteristic chains, initiated by Cauchy, which shows well the nature of the relations implied by the given differential equation; the alternative ways of carrying out the necessary integrations are suggested by considering the method of Jacobi and Mayer, while a good summary is obtained by the formulation in terms of a Pfaffian expression.

Characteristic chains.

Consider a solution of F = 0 expressed by the three independent
equations F = 0, G = 0, H = 0. If it be a solution in which there is
more than one relation connecting x, y, z, let new variables x', y',
z', p', q' be introduced, as before explained under PFAFFIAN
EXPRESSIONS, in which z' is of the form

z' = z - p1x1 - ... - p_s x_s (s = 1 or 2),

so that the solution becomes of a form z' = [psi](x'y'), p' =
d[psi]/dx', q' = d[psi]/dy', which then will identically satisfy the
transformed equations F' = 0, G' = 0, H' = 0. The equation F' = 0, if
x', y', z' be regarded as fixed, states that the plane Z - z' = p'(X -
x') + q'(Y - y') is tangent to a certain cone whose vertex is (x', y',
z'), the consecutive point (x' + dx', y' + dy', z' + dz') of the
generator of contact being such that

/dF' /dF' / / dF' dF'\
dx'/ -- = dy'/ -- = dz'/ ( p'--- + q' --- ).
/ dp' / dq' / \ dp' dq'/

Passing in this direction on the surface z' = [psi](x', y') the
tangent plane of the surface at this consecutive point is (p' + dp',
q' + dq'), where, since F'(x', y', [psi], d[psi]/dx', d[psi]/dy') = 0
is identical, we have dx' (dF'/dx' + p'dF'/dz') + dp'dF'/dp' = 0. Thus
the equations, which we shall call the characteristic equations,

/dF' /dF' // dF' dF'\ // dF' dF'\
dx'/ --- = dy'/ --- = dz'/( p' --- + q'--- ) = dp'/( - --- - p'--- )
/ dp' / dq' / \ dp' dq'/ / \ dx' dz'/

// dF' dF'\
= dq'/( - --- - q'--- )
/ \ dy' dz'/

are satisfied along a connectivity of [oo]^1 elements consisting of a
curve on z' = [psi](x', y') and the tangent planes of the surface
along this curve. The equation F' = 0, when p', q' are fixed,
represents a curve in the plane Z - z' = p'(X - x') + q'(Y - y')
passing through (x', y', z'); if (x' + [delta]x', y' + [delta]y', z' +
[delta]z') be a consecutive point of this curve, we find at once

/dF' dF'\ /dF' dF'\
[delta]x'( --- + p'--- ) + [delta]y'( --- + q'--- ) = 0;
\dx' dz'/ \dy' dz'/

thus the equations above give [delta]x'dp' + [delta]y'dq' = 0, or the
tangent line of the plane curve, is, on the surface z' = [psi](x',
y'), in a direction conjugate to that of the generator of the cone.
Putting each of the fractions in the characteristic equations equal to
dt, the equations enable us, starting from an arbitrary element x'0,
y'0, z'0, p'0, q'0, about which all the quantities F', dF'/dp', &c.,
occurring in the denominators, are developable, to define, from the
differential equation F' = 0 alone, a connectivity of [oo]^1 elements,
which we call a _characteristic chain_; and it is remarkable that when
we transform again to the original variables (x, y, z, p, q), the form
of the differential equations for the chain is unaltered, so that they
can be written down at once from the equation F = 0. Thus we have
proved that the characteristic chain starting from any ordinary
element of any integral of this equation F = 0 consists only of
elements belonging to this integral. For instance, if the equation do
not contain p, q, the characteristic chain, starting from an arbitrary
plane through an arbitrary point of the surface F = 0, consists of a
pencil of planes whose axis is a tangent line of the surface F = 0. Or
if F = 0 be of the form Pp + Qq = R, the chain consists of a curve
satisfying dx/P = dy/Q = dz/R and a single infinity of tangent planes
of this curve, determined by the tangent plane chosen at the initial
point. In all cases there are [oo]^3 characteristic chains, whose
aggregate may therefore be expected to exhaust the [oo]^4 elements
satisfying F = 0.

Complete integral constructed with characteristic chains.

Consider, in fact, a single infinity of connected elements each
satisfying F = 0, say a chain connectivity T, consisting of elements
specified by x0, y0, z0, p0, q0, which we suppose expressed as
functions of a parameter u, so that

U0 = dz0/du - p0dx0/du - q0dy0/du

is everywhere zero on this chain; further, suppose that each of F,
dF/dp, ... , dF/dx + pdF/dz is developable about each element of this
chain T, and that T is _not_ a characteristic chain. Then consider the
aggregate of the characteristic chains issuing from all the elements
of T. The [oo]^2 elements, consisting of the aggregate of these
characteristic chains, satisfy F = 0, provided the chain connectivity
T consists of elements satisfying F = 0; for each characteristic chain
satisfies dF = 0. It can be shown that these chains are connected; in
other words, that if x, y, z, p, q, be any element of one of these
characteristic chains, not only is

dz/dt - pdx/dt - qdy/dt = 0,

as we know, but also U = dz/du - pdx/du - qdy/du is also zero. For we
have

dU d /dz dx dy\ d /dz dx dy\
-- = --( -- - p-- - q-- ) - --( -- - p-- - q-- )
dt dt \du du du/ du \dt dt dt/

dp dx dp dx dq dy dq dy
= -- -- - -- -- + -- -- - -- -- ,
du dt dt du du dt dt du

which is equal to

dp dF dx /dF dF\ dq dF dy /dF dF\ dF
-- -- + --( -- + p-- ) + -- -- + --( -- + q-- ) = - -- U.
du dp du \dx dz/ du dq du \dy dz/ dz

dF
As -- is a developable function of t, this, giving
dz
_
/ / t dF \
U = U_{0} exp( - | --dt ),
\ _/t0 dz /

shows that U is everywhere zero. Thus integrals of F = 0 are
obtainable by considering the aggregate of characteristic chains
issuing from arbitrary chain connectivities T satisfying F = 0; and
such connectivities T are, it is seen at once, determinable without
integration. Conversely, as such a chain connectivity T can be taken
out from the elements of any given integral all possible integrals are
obtainable in this way. For instance, an arbitrary curve in space,
given by x0 = [theta](u), y0 = [phi](u), z0 = [psi](u), determines by
the two equations F(x0, y0, z0, p0, q0) = 0, [psi]'(u) = p0[theta]'(u)
+ q0[phi]'(u), such a chain connectivity T, through which there passes
a perfectly definite integral of the equation F = 0. By taking [oo]^2
initial chain connectivities T, as for instance by taking the curves
x0 = [theta], y0 = [phi], z0 = [psi] to be the [oo]^2 curves upon an
arbitrary surface, we thus obtain [oo]^2 integrals, and so [oo]^4
elements satisfying F = 0. In general, if functions G, H, independent
of F, be obtained, such that the equations F = 0, G = b, H = c
represent an integral for all values of the constants b, c, these
equations are said to constitute a _complete integral_. Then [oo]^4
elements satisfying F = 0 are known, and in fact every other form of
integral can be obtained without further integrations.

Operations necessary for integration of F = a.

In the foregoing discussion of the differential equations of a
characteristic chain, the denominators dF/dp, ... may be supposed to
be modified in form by means of F = 0 in any way conducive to a simple
integration. In the immediately following explanation of ideas,
however, we consider indifferently all equations F = constant; when a
function of x, y, z, p, q is said to be zero, it is meant that this is
so identically, not in virtue of F = 0; in other words, we consider
the integration of F = a, where a is an arbitrary constant. In the
theory of linear partial equations we have seen that the integration
of the equations of the characteristic chains, from which, as has just
been seen, that of the equation F = a follows at once, would be
involved in completely integrating the single linear homogeneous
partial differential equation of the first order [Ff] = 0 where the
notation is that explained above under CONTACT TRANSFORMATIONS. One
obvious integral is f = F. Putting F = a, where a is arbitrary, and
eliminating one of the independent variables, we can reduce this
equation [Ff] = 0 to one in four variables; and so on. Calling, then,
the determination of a single integral of a single homogeneous partial
differential equation of the first order in n independent variables,
_an operation of order_ n - 1, the characteristic chains, and
therefore the most general integral of F = a, can be obtained by
successive operations of orders 3, 2, 1. If, however, an integral of F
= a be represented by F = a, G = b, H = c, where b and c are arbitrary
constants, the expression of the fact that a characteristic chain of F
= a satisfies dG = 0, gives [FG] = 0; similarly, [FH] = 0 and [GH] =
0, these three relations being identically true. Conversely, suppose
that an integral G, independent of F, has been obtained of the
equation [Ff] = 0, which is an operation of order three. Then it
follows from the identity [f[[phi][psi]]] + [[phi][[psi]f]] +
[[psi][f[phi]]] = df/dz [[psi][phi]] + d[phi]/dz [psif] + d[psi]/dz
[f[phi]] before remarked, by putting [phi] = F, [psi] = G, and then
[Ff] = A(f), [Gf] = B(f), that AB(f) - BA(f) = dF/dz B(f) - dG/dz
A(f), so that the two linear equations [Ff] = 0, [Gf] = 0 form a
complete system; as two integrals F, G are known, they have a common
integral H, independent of F, G, determinable by an operation of order
one only. The three functions F, G, H thus identically satisfy the
relations [FG] = [GH] = [FH] = 0. The [oo]^2 elements satisfying F = a,
G = b, H = c, wherein a, b, c are assigned constants, can then be seen
to constitute an integral of F = a. For the conditions that a
characteristic chain of G = b issuing from an element satisfying F =
a, G = b, H = c should consist only of elements satisfying these three
equations are simply [FG] = 0, [GH] = 0. Thus, starting from an
arbitrary element of (F = a, G = b, H = c), we can single out a
connectivity of elements of (F = a, G = b, H = c) forming a
characteristic chain of G = b; then the aggregate of the
characteristic chains of F = a issuing from the elements of this
characteristic chain of G = b will be a connectivity consisting only
of elements of

(F = a, G = b, H = c),

and will therefore constitute an integral of F = a; further, it will
include all elements of (F = a, G = b, H = c). This result follows
also from a theorem given under CONTACT TRANSFORMATIONS, which shows,
moreover, that though the characteristic chains of F = a are not
determined by the three equations F = a, G = b, H = c, no further
integration is now necessary to find them. By this theorem, since
identically [FG] = [GH] = [FH] = 0, we can find, by the solution of
linear algebraic equations only, a non-vanishing function [sigma] and
two functions A, C, such that

dG - AdF - CdH = [sigma](dz - pdz - qdy);

thus all the elements satisfying F = a, G = b, H = c, satisfy dz = pdx
+ qdy and constitute a connectivity, which is therefore an integral of
F = a. While, further, from the associated theorems, F, G, H, A, C are
independent functions and [FC] = 0. Thus C may be taken to be the
remaining integral independent of G, H, of the equation [Ff] = 0,
whereby the characteristic chains are entirely determined.

The single equation F = 0 and Pfaffian formulations.

When we consider the particular equation F = 0, neglecting the case
when neither p nor q enters, and supposing p to enter, we may express
p from F = 0 in terms of x, y, z, q, and then eliminate it from all
other equations. Then instead of the equation [Ff] = 0, we have, if F
= 0 give p = [psi](x, y, z, q), the equation

/df df\ d[psi] /df df\ /d[psi] d[psi]\ df
[Sigma]f = - ( -- + [psi] -- ) + ------ ( -- + q -- ) - ( ------ + q ------ ) -- = 0,
\dx dz/ dq \dy dz/ \ dy dz / dq

moreover obtainable by omitting the term in df/dp in [p-[psi], f] = 0.
Let x0, y0, z0, q0, be values about which the coefficients in this
equation are developable, and let [zeta], [eta], [omega] be the
principal solutions reducing respectively to z, y and q when x = x0.
Then the equations p = [psi], [zeta] = z0, [eta] = y0, [omega] = q0
represent a characteristic chain issuing from the element x0, y0, z0,
[psi]0, q0; we have seen that the aggregate of such chains issuing
from the elements of an arbitrary chain satisfying

dz0 = p0dx0 - q0dy0 = 0

constitute an integral of the equation p = [psi]. Let this arbitrary
chain be taken so that x0 is constant; then the condition for initial
values is only

dz0 - q0dy0 = 0,

and the elements of the integral constituted by the characteristic
chains issuing therefrom satisfy

d[zeta] - [omega]d[eta] = 0.

Hence this equation involves dz - [psi]dx - qdy = 0, or we have

dz - [psi]dx - qdy = [sigma](d[zeta] - [omega]d[eta]),

where [sigma] is not zero. Conversely, the integration of p = [psi]
is, essentially, the problem of writing the expression dz - [psi]dx -
qdy in the form [sigma](d[zeta] - [omega]d[eta]), as must be possible
(from what was said under _Pfaffian Expressions_).

System of equations of the first order.

To integrate a system of simultaneous equations of the first order X1
= a1, ... Xr = ar in n independent variables x1, ... xn and one
dependent variable z, we write p1 for dz/dx1, &c., and attempt to find
n + 1 - r further functions Z, X_r+1 ... Xn, such that the equations Z
= a, Xi = ai,(i = 1, ... n) involve dz - p1dx1 - ... - pndxn = 0. By
an argument already given, the common integral, if existent, must be
satisfied by the equations of the characteristic chains of any one
equation Xi = ai; thus each of the expressions [Xi Xj] must vanish in
virtue of the equations expressing the integral, and we may without
loss of generality assume that each of the corresponding 1/2r(r - 1)
expressions formed from the r given differential equations vanishes in
virtue of these equations. The determination of the remaining n + 1 -
r functions may, as before, be made to depend on characteristic
chains, which in this case, however, are manifolds of r dimensions
obtained by integrating the equations [X1f] = 0, ... [Xrf] = 0; or
having obtained one integral of this system other than X1, ... Xr, say
Xr+1, we may consider the system [X1f] = 0, ... [X_r+1 f] = 0, for
which, again, we have a choice; and at any stage we may use Mayer's
method and reduce the simultaneous linear equations to one equation
involving parameters; while if at any stage of the process we find
some but not all of the integrals of the simultaneous system, they can
be used to simplify the remaining work; this can only be clearly
explained in connexion with the theory of so-called function groups
for which we have no space. One result arising is that the
simultaneous system p1 = [phi]1, ... pr = [phi]r, wherein p1, ... pr
are not involved in [phi]1, ... [phi]r, if it satisfies the 1/2r(r - 1)
relations [pi - [phi]i, pj - [phi]j] = 0, has a solution z = [psi](x1,
... xn), p1 = d[psi]/dx1, ... pn = d[psi]/dxn, reducing to an
arbitrary function of x_r+1, ... xn only, when x1 = x1^0, ... xr =
xr^0 under certain conditions as to developability; a generalization
of the theorem for linear equations. The problem of integration of
this system is, as before, to put

dz - [phi]1dx1 - ... - [phi]_r dx_r - p_r+1 dx_r+1 - ... - p_n dx_n

into the form [sigma](d[zeta] - [omega]_r+1 + d[xi]_r+1 - ... -
[omega]_n d[xi]_n); and here [zeta], [xi]_r+1, ... [xi]_n,
[omega]_r+1, ... [omega]_n may be taken, as before, to be principal
integrals of a certain complete system of linear equations; those,
namely, determining the characteristic chains.

Equations of dynamics.

If L be a function of t and of the 2n quantities x1, ... xn, [.x]1,
... [.x]n, where [.x]i, denotes dxi/dt, &c., and if in the n equations

d / dL \ dL
--- (--------) = ----
dt \ dx_i / dx_i

we put p_i = dL/d[.x]_i, and so express [.x]1 , ... [.x]_n in terms of
t, x_i, ... x_n, p1, ... p_n, assuming that the determinant of the
quantities d^2L/dx_i d[.x]_j is not zero; if, further, H denote the
function of t, x1, ... xn, p1, ... pn, numerically equal to p1[.x]1 +
... + pn[.x]n - L, it is easy to prove that dpi/dt = -dH/dxi, dxi/dt =
dH/dp_i. These so-called _canonical_ equations form part of those for
the characteristic chains of the single partial equation dz/dt + H(t,
x1, ... xn, dz/dx1, ..., dz/dx_n) = 0, to which then the solution of
the original equations for x1 ... xn can be reduced. It may be shown
(1) that if z = [psi](t, x1, ... xn, c1, .. cn) + c be a complete
integral of this equation, then pi = d[psi]/dx_i, d[psi]/dc_i = e_i are
2n equations giving the solution of the canonical equations referred
to, where c1 ... cn and e1, ... en are arbitrary constants; (2) that
if xi = Xi(t, x^01, ... pn^0), pi=Pi(t, x1^0, ... p^0n) be the
principal solutions of the canonical equations for t = t^0, and
[omega] denote the result of substituting these values in p1dH/dp1 +
... + pndH/dpn - H, and [Omega] = [int] [t0 to t] [omega]dt, where,
after integration, [Omega] is to be expressed as a function of t, x1,
... xn, x1^0, ... xn^0, then z = [Omega] + z^0 is a complete integral
of the partial equation.

Application of theory of continuous groups to formal theories.

A system of differential equations is said to allow a certain continuous group of transformations (see GROUPS, THEORY OF) when the introduction for the variables in the differential equations of the new variables given by the equations of the group leads, for all values of the parameters of the group, to the same differential equations in the new variables. It would be interesting to verify in examples that this is the case in at least the majority of the differential equations which are known to be integrable in finite terms. We give a theorem of very general application for the case of a simultaneous complete system of linear partial homogeneous differential equations of the first order, to the solution of which the various differential equations discussed have been reduced. It will be enough to consider whether the given differential equations allow the infinitesimal transformations of the group.

It can be shown easily that sufficient conditions in order that a
complete system [Pi]1f = 0 ... [Pi]kf = 0, in n independent variables,
should allow the infinitesimal transformation Pf = 0 are expressed by
k equations [Pi]_i Pf - P[Pi]_i f = [lambda]_i1 [Pi]1f + ... +
[lambda]_ik [Pi]_kf. Suppose now a complete system of n - r equations
in n variables to allow a group of r infinitesimal transformations
(P1f, ..., Prf) which has an invariant subgroup of r - 1 parameters
(P1f, ..., Pr-1f), it being supposed that the n quantities [Pi]1f,
..., [Pi]_n-r f, P1 f, ..., P_r f are not connected by an identical
linear equation (with coefficients even depending on the independent
variables). Then it can be shown that one solution of the complete
system is determinable by a quadrature. For each of [Pi]_i P_[sigma] f
- P_[sigma] [Pi]_i f is a linear function of [Pi]1f, ..., [Pi]_n-r f
and the simultaneous system of independent equations [Pi]1f = 0, ...
[Pi]_n-r f = 0, P1f = 0, ... P_r-1 f = 0 is therefore a complete
system, allowing the infinitesimal transformation Prf. This complete
system of n - 1 equations has therefore one common solution [omega],
and P_r([omega]) is a function of [omega]. By choosing [omega]
suitably, we can then make Pr([omega]) = 1. From this equation and the
n - 1 equations [Pi]_i[omega] = 0, P_[sigma][omega] = 0, we can
determine [omega] by a quadrature only. Hence can be deduced a much
more general result, _that if the group of r parameters be integrable,
the complete system can be entirety solved by quadratures_; it is only
necessary to introduce the solution found by the first quadrature as
an independent variable, whereby we obtain a complete system of n - r
equations in n - 1 variables, subject to an integrable group of r - 1
parameters, and to continue this process. We give some examples of the
application of the theorem. (1) If an equation of the first order y' =
[psi](x, y) allow the infinitesimal transformation [xi]df/dx +
[eta]df/dy, the integral curves [omega](x, y) = y^0, wherein [omega](x,
y) is the solution of df/dx + [psi](x, y) df/dy = 0 reducing to y for
x = x^0, are interchanged among themselves by the infinitesimal
transformation, or [omega](x, y) can be chosen to make [xi]d[omega]/dx
+ [eta]d[omega]/dy = 1; this, with d[omega]/dx + [psi]d[omega]/dy = 0,
determines [omega] as the integral of the complete differential (dy -
[psi]dx)/([eta] - [psi][xi]). This result itself shows that every
ordinary differential equation of the first order is subject to an
infinite number of infinitesimal transformations. But every
infinitesimal transformation [xi]df/dx + [eta]df/dy can by change of
variables (after integration) be brought to the form df/dy, and all
differential equations of the first order allowing this group can then
be reduced to the form F(x, dy/dx) = 0. (2) In an ordinary equation of
the second order y" = [psi](x, y, y'), equivalent to dy/dx = y1,
dy1/dx = [psi](x, y, y1), if H, H1 be the solutions for y and y1
chosen to reduce to y^0 and y1^0 when x = x^0, and the equations H = y,
H1= y1 be equivalent to [omega] = y^0, [omega]1 = y1^0, then [omega],
[omega]1 are the principal solutions of [Pi]f = df/dx + y1df/dy +
[psi]df/dy1 = 0. If the original equation allow an infinitesimal
transformation whose first _extended_ form (see GROUPS) is Pf =
[xi]df/dx + [eta]df/dy + [eta]1df/dy1, where [eta]1[delta]t is the
increment of dy/dx when [xi][delta]t, [eta][delta]t are the increments
of x, y, and is to be expressed in terms of x, y, y1, then each of
P[omega] and P[omega]1 must be functions of [omega] and [omega]1, or
the partial differential equation [Pi]f must allow the group Pf. Thus
by our general theorem, if the differential equation allow a group of
two parameters (and such a group is always integrable), it can be
solved by quadratures, our explanation sufficing, however, only
provided the form [Pi]f and the two infinitesimal transformations are
not linearly connected. It can be shown, from the fact that [eta]1 is
a quadratic polynomial in y1, that no differential equation of the
second order can allow more than 8 really independent infinitesimal
transformations, and that every homogeneous linear differential
equation of the second order allows just 8, being in fact reducible to
d^2y/dx^2 = 0. Since every group of more than two parameters has
subgroups of two parameters, a differential equation of the second
order allowing a group of more than two parameters can, as a rule, be
solved by quadratures. By transforming the group we see that if a
differential equation of the second order allows a single
infinitesimal transformation, it can be transformed to the form F(x,
d[gamma]/dx, d^2[gamma]/dx^2); this is not the case for every
differential equation of the second order. (3) For an ordinary
differential equation of the third order, allowing an integrable group
of three parameters whose infinitesimal transformations are not
linearly connected with the partial equation to which the solution of
the given ordinary equation is reducible, the similar result follows
that it can be integrated by quadratures. But if the group of three
parameters be simple, this result must be replaced by the statement
that the integration is reducible to quadratures and that of a
so-called Riccati equation of the first order, of the form dy/dx = A +
By + Cy^2, where A, B, C are functions of x. (4) Similarly for the
integration by quadratures of an ordinary equation yn = [psi](x, y,
y1, ... yn-1) of any order. Moreover, the group allowed by the
equation may quite well consist of extended contact transformations.
An important application is to the case where the differential
equation is the resolvent equation defining the group of
transformations or rationality group of another differential equation
(see below); in particular, when the rationality group of an ordinary
linear differential equation is integrable, the equation can be solved
by quadratures.

Consideration of function theories of differential equations.

Following the practical and provisional division of theories of differential equations, to which we alluded at starting, into transformation theories and function theories, we pass now to give some account of the latter. These are both a necessary logical complement of the former, and the only remaining resource when the expedients of the former have been exhausted. While in the former investigations we have dealt only with values of the independent variables about which the functions are developable, the leading idea now becomes, as was long ago remarked by G. Green, the consideration of the neighbourhood of the values of the variables for which this developable character ceases. Beginning, as before, with existence theorems applicable for ordinary values of the variables, we are to consider the cases of failure of such theorems.

A general existence theorem.

When in a given set of differential equations the number of equations is greater than the number of dependent variables, the equations cannot be expected to have common solutions unless certain conditions of compatibility, obtainable by equating different forms of the same differential coefficients deducible from the equations, are satisfied. We have had examples in systems of linear equations, and in the case of a set of equations p1 = [phi]1, ..., pr = [phi]r. For the case when the number of equations is the same as that of dependent variables, the following is a general theorem which should be referred to: Let there be r equations in r dependent variables z1, ... zr and n independent variables x1, ... xn; let the differential coefficient of z[sigma] of highest order which enters be of order h[sigma], and suppose d^h_[sigma] z_[sigma]/dx1^h_[sigma] to enter, so that the equations can be written d^h_[sigma] z_[sigma]/dx1^h_[sigma] = [Phi]_[sigma], where in the general differential coefficient of z_[rho] which enters in [Phi]_[sigma], say

d^(k1 + ... + kn) z_[rho]/dx1^k1 ... dx_n^k_n,

we have k1 < h_[rho] and k1 + ... + k_n <= h_[rho]. Let a1, ... an, b1, ... br, and b[rho]_(k1 ... kn) be a set of values of

x1, ... x_n, z1, ... z_r

and of the differential coefficients entering in [Phi]_[sigma] about which all the functions [Phi]1, ... [Phi]_r, are developable. Corresponding to each dependent variable z_[sigma], we take now a set of h_[sigma] functions of x2, ... xn, say [phi][sigma], [phi][sigma]^(1), ..., [phi][sigma]^(h-1) arbitrary save that they must be developable about a2, a3, ... an, and such that for these values of x2, ... xn, the function [phi]_[rho] reduces to b_[rho], and the differential coefficient

d^(k2 + ... + kn) [phi]_[rho]^(k1)/dx2^k2 ... dx_n^kn

reduces to b^kn_(k1 ... kn). Then the theorem is that there exists one, and only one, set of functions z1, ... z_r, of x2, ... x_n developable about a1, ... an satisfying the given differential equations, and such that for x1 = a1 we have

z_[sigma] = [phi]_[sigma], dz_[sigma]/dx1 = [phi]_[sigma]^(1), ...
d^(h_[sigma]-1) z_[sigma]/d^(h_[sigma]-1) x1 = [phi][sigma]^(h_[sigma]-1).

And, moreover, if the arbitrary functions [phi]_[sigma], [phi]_[sigma]^(1) ... contain a certain number of arbitrary variables t1, ... tm, and be developable about the values t1^0, ... tm^0 of these variables, the solutions z1, ... zr will contain t1, ... tm, and be developable about t1^0, ... tm^0.

Singular points of solutions.

The proof of this theorem may be given by showing that if ordinary
power series in x1 - -a1, ... xn - an, t1 - t1^0, ... tm - tm^0 be
substituted in the equations wherein in z[sigma] the coefficients of
(x1 - a1)^0, x1 - a1, ..., (x1 - a1)^(h_[sigma]-1) are the arbitrary
functions [phi]_[sigma], [phi]_[sigma]^(1), ..., [phi]_[sigma]^h-1,
divided respectively by 1, 1!, 2!, &c., then the differential
equations determine uniquely all the other coefficients, and that the
resulting series are convergent. We rely, in fact, upon the theory of
monogenic analytical functions (see FUNCTION), a function being
determined entirely by its development in the neighbourhood of one set
of values of the independent variables, from which all its other
values arise by _continuation_; it being of course understood that the
coefficients in the differential equations are to be continued at the
same time. But it is to be remarked that there is no ground for
believing, if this method of continuation be utilized, that the
function is single-valued; we may quite well return to the same values
of the independent variables with a different value of the function;
belonging, as we say, to a different branch of the function; and there
is even no reason for assuming that the number of branches is finite,
or that different branches have the same singular points and regions
of existence. Moreover, and this is the most difficult consideration
of all, all these circumstances may be dependent upon the values
supposed given to the arbitrary constants of the integral; in other
words, the singular points may be either _fixed_, being determined by
the differential equations themselves, or they may be _movable_ with
the variation of the arbitrary constants of integration. Such
difficulties arise even in establishing the reversion of an elliptic
integral, in solving the equation

/dx\^2
( -- ) = (x-a1)(x - a2)(x - a3)(x - a4);
\ds/

about an ordinary value the right side is developable; if we put x -
a1 = t1^2, the right side becomes developable about t1 = 0; if we put x
= 1/t, the right side of the changed equation is developable about t =
0; it is quite easy to show that the integral reducing to a definite
value x0 for a value s0 is obtainable by a series in integral powers;
this, however, must be supplemented by showing that for no value of s
does the value of x become entirely undetermined.

Linear differential equations with rational coefficients.

These remarks will show the place of the theory now to be sketched of
a particular class of ordinary linear homogeneous differential
equations whose importance arises from the completeness and generality
with which they can be discussed. We have seen that if in the
equations dy/dx = y1, dy1/dx = y2, ..., dy_n-2/dx = y_n-1,

dy_n-1/dx = a_n y + a_n-1 y1 + ... + a1 y_n-1,

where a1, a2, ..., an are now to be taken to be rational functions of
x, the value x = x^0 be one for which no one of these rational
functions is infinite, and y^0, y^01, ..., y^0_n-1 be quite arbitrary
finite values, then the equations are satisfied by

y = y^0u + y^01u1 + ... + y^0_n-1 u_n-1,

where u, u1, ..., un-1 are functions of x, independent of y^0, ...
y^0_n-1, developable about x = x^0; this value of y is such that for x =
x^0 the functions y, y1 ... y_n-1 reduce respectively to y^0, y^01, ...
y^0_n-1; it can be proved that the region of existence of these series
extends within a circle centre x^0 and radius equal to the distance
from x^0 of the nearest point at which one of a1, ... an becomes
infinite. Now consider a region enclosing x^0 and only one of the
places, say [Sigma], at which one of a1, ... an becomes infinite. When
x is made to describe a closed curve in this region, including this
point [Sigma] in its interior, it may well happen that the
continuations of the functions u, u1, ..., u_n-1 give, when we have
returned to the point x, values v, v1, ..., v_n-1, so that the
integral under consideration becomes changed to y^0 + y^01v1 + ... +
y^0_n-1 v_n-1. At x^0 let this branch and the corresponding values of
y1, ... y_n-1 be [eta]^0, [eta]^0_1, ... [eta]^0_n-1; then, as there is
only one series satisfying the equation and reducing to ([eta]^0,
[eta]^0_1, ... [eta]^0_n-1) for x = x^0 and the coefficients in the
differential equation are single-valued functions, we must have
[eta]^0_u + [eta]^0_1u1 + ... + [eta]^0_n-1 u_n-1 = y^0v + y^01v1 + ... +
y^0_n-1 v_n-1; as this holds for arbitrary values of y^0 ... y^0_n-1,
upon which u, ... u_n-1 and v, ... v_n-1 do not depend, it follows
that each of v, ... v_n-1 is a linear function of u, ... u_n-1 with
constant coefficients, say v_i = A_i1 u + ... + A_in u_n-1. Then

y^0v + ... + y^0_n-1 v_n-1 = ([Sigma]_i A_i1 y_i^0)u + ... + ([Sigma]_i A_in y^0_i)u_n-1;

this is equal to [mu](y^0u + ... + y^0_n-1 u_n-1) if [Sigma]_i A_ir y^0_i
= [mu]y^0_r-1; eliminating y^0 ... y^0_n-1 from these linear equations,
we have a determinantal equation of order n for [mu]; let [mu]1 be one
of its roots; determining the ratios of y^0, y1^0, ... y^0_n-1 to satisfy
the linear equations, we have thus proved that there exists an
integral, H, of the equation, which when continued round the point
[Sigma] and back to the starting-point, becomes changed to H1 =
[mu]1H. Let now [xi] be the value of x at [Sigma] and r1 one of the
values of (1/2[pi]i) log [mu]1; consider the function (x - [xi])^r1 H;
when x makes a circuit round x = [xi], this becomes changed to

exp(-2[pi]ir1) (x - [xi])^-r1 [mu]H,

that is, is unchanged; thus we may put H = (x - [xi])^r1 [phi]1,
[phi]1 being a function single-valued for paths in the region
considered described about [Sigma], and therefore, by Laurent's
Theorem (see FUNCTION), capable of expression in the annular region
about this point by a series of positive and negative integral powers
of x - [xi], which in general may contain an infinite number of
negative powers; there is, however, no reason to suppose r1 to be an
integer, or even real. Thus, if all the roots of the determinantal
equation in [mu] are different, we obtain n integrals of the forms (x
-[xi])^r1 phi1, ..., (x - [xi])^rn [phi]_n. In general we obtain as
many integrals of this form as there are really different roots; and
the problem arises to discover, in case a root be k times repeated, k
- 1 equations of as simple a form as possible to replace the k - 1
equations of the form y^0 + ... + y^0_n-1 v_n-1 = [mu](y^0 + ... + y^0_n-1
u_n-1) which would have existed had the roots been different. The most
natural method of obtaining a suggestion lies probably in remarking
that if r2 = r1 + h, there is an integral [(x - [xi])^(r1 + h) [phi]2
- (x -[xi])^r1 [phi]1]/h, where the coefficients in [phi]2 are the
same functions of r1 + h as are the coefficients in [phi]1 of r1; when
h vanishes, this integral takes the form
_ _
| d[phi]1 |
(x - [xi])^r1 | ------- + [phi]1 log (x - [xi])|,
|_ dr1 _|

or say (x-[xi])^r1 [[phi]1 + [psi]1 log (x - [xi])];

denoting this by 2[pi]i[mu]1K, and (x-[xi])^r1 [phi]1 by H, a circuit
of the point [xi] changes K into

1
K' = ----------- [e^(2[pi]ir1) (x - [xi])^r1 [psi]1 + e^(2[pi]ir1) (x - [xi])^r1 [phi]1 (2[pi]i + log(x - [xi]))]
2[pi]i[mu]1

= [mu]1K + H.

A similar artifice suggests itself when three of the roots of the
determinantal equation are the same, and so on. We are thus led to the
result, which is justified by an examination of the algebraic
conditions, that whatever may be the circumstances as to the roots of
the determinantal equation, n integrals exist, breaking up into
batches, the values of the constituents H1, H2, ... of a batch after
circuit about x = [xi] being H1' = [mu]1H1, H2' = [mu]1H2 + H1, H3' =
[mu]1H3 + H2, and so on. And this is found to lead to the forms (x -
[xi])^r1 [phi]1, (x - [xi])^r1 [[psi]1 + [phi]1 log (x - [xi])], (x -
[xi])^r1 [[chi]1 + [chi]2 log (x - [xi]) + [phi]1(log(x - [xi]))^2],
and so on. Here each of [phi]1, [psi]1, [chi]1, [chi]2, ... is a
series of positive and negative integral powers of x - [xi] in which
the number of negative powers may be infinite.

Regular equations.

It appears natural enough now to inquire whether, under proper
conditions for the forms of the rational functions a1, ... an, it may
be possible to ensure that in each of the series [phi]1, [psi]1,
[chi]1, ... the number of negative powers shall be finite. Herein
lies, in fact, the limitation which experience has shown to be
justified by the completeness of the results obtained. Assuming n
integrals in which in each of [phi]1, [psi]1, [chi]1 ... the number of
negative powers is finite, there is a definite homogeneous linear
differential equation having these integrals; this is found by forming
it to have the form

y'^n = (x - [xi])^-1 b1y'^(n-1) + (x - [xi])^-2 b2y'^(n-2) + ... +(x - [xi])^-n b_n y,

where b1, ... bn are finite for x = [xi]. Conversely, assume the
equation to have this form. Then on substituting a series of the form
(x - [xi])^r [1 + A1(x - [xi]) + A2(x - [xi])^2 + ... ] and equating
the coefficients of like powers of x-[xi], it is found that r must be
a root of an algebraic equation of order n; this equation, which we
shall call the index equation, can be obtained at once by substituting
for y only (x - [xi])^r and replacing each of b1, ... bn by their
values at x = [xi]; arrange the roots r1, r2, ... of this equation so
that the real part of ri is equal to, or greater than, the real part
of r_i+1, and take r equal to r1; it is found that the coefficients
A1, A2 ... are uniquely determinate, and that the series converges
within a circle about x = [xi] which includes no other of the points
at which the rational functions a1 ... an become infinite. We have
thus a solution H1 = (x -[xi])^r1 [phi]1 of the differential equation.
If we now substitute in the equation y = H1 f[eta]dx, it is found to
reduce to an equation of order n - 1 for [eta] of the form

[eta]'^(n-1) = (x - [xi])^-1 c1[eta]'^(n-2) + ... + (x-[xi])^(n-1) c_n-1 [eta],

where c1, ... c_n-1 are not infinite at x = [xi]. To this equation
precisely similar reasoning can then be applied; its index equation
has in fact the roots r2 - r1 - 1, ... , rn - r1 - 1; if r2 - r1 be
zero, the integral (x - [xi])^-1 [psi]1 of the [eta] equation will
give an integral of the original equation containing log (x - [xi]);
if r2 - r1 be an integer, and therefore a negative integer, the same
will be true, unless in [psi]1 the term in (x - [xi])^(r1 - r2) be
absent; if neither of these arise, the original equation will have an
integral (x -[xi])^r2 [phi]2. The [eta] equation can now, by means of
the one integral of it belonging to the index r2 - r1 - 1, be
similarly reduced to one of order n - 2, and so on. The result will be
that stated above. We shall say that an equation of the form in
question is _regular_ about x = [xi].

Fuchsian equations.

Equation of the second order.