possesses an integral P0 + tP1 + t^2P2/2! + ... in which the
coefficients in P0 are real and positive, and each not less than the
absolute value of the corresponding coefficient in p0. If a, b be
developable for x, y both in absolute value less than r and for t less
in absolute value than R, and for such values a, b be both less in
absolute value than the real positive constant M, it is not difficult
to verify that we may take
/ x + y\-1 / t\-1
A = B = M( 1 - ----- ) ( 1 - - ),
\ r / \ R/
and obtain
_ _
| 4MR / x + y\-2 / t\-1 |1/2
F = r - (r - x - y) | 1 - ---(1 - ------) log (1 - - ) |,
|_ r \ r / \ R/ _|
and that this solves the problem when x, y, t are sufficiently small
for the two cases p0 = x, p0 = y. One obvious application of the
general theorem is to the proof of the existence of an integral of an
ordinary linear differential equation given by the n equations dy/dx =
y1, dy1/dx = y2, ...,
dy_n-1/dx = p - p1 y_n-1 - ... - p_n y;
but in fact any simultaneous system of ordinary equations is reducible
to a system of the form
dx1/dt = [phi](tx1, ... x_n).
Simultaneous linear partial equations.
Complete systems of linear partial equations.
Jacobian systems.
Suppose we have k homogeneous linear partial equations of the first
order in n independent variables, the general equation being
a_[sigma]1 df/dx1 + ... + a_[sigma]n df/dx_n = 0, where [sigma] = 1,
... k, and that we desire to know whether the equations have common
solutions, and if so, how many. It is to be understood that the
equations are linearly independent, which implies that k <= n and not
every determinant of k rows and columns is identically zero in the
matrix in which the i-th element of the [sigma]-th row is a[sigma]_i(i
= 1, ... n, [sigma] = 1, ... k). Denoting the left side of the
[sigma]-th equation by P[sigma]f, it is clear that every common
solution of the two equations P_[sigma]f = 0, P_[rho]f = 0, is also a
solution of the equation P_[rho](P_[sigma]f), P_[sigma](P_[rho]f), We
immediately find, however, that this is also a linear equation,
namely, [Sigma]H_i df/dx_i = 0 where H_i = P[rho]a[sigma]_i -
P[sigma]a[rho]_i, and if it be not already contained among the given
equations, or be linearly deducible from them, it may be added to
them, as not introducing any additional limitation of the possibility
of their having common solutions. Proceeding thus with every pair of
the original equations, and then with every pair of the possibly
augmented system so obtained, and so on continually, we shall arrive
at a system of equations, linearly independent of each other and
therefore not more than n in number, such that the combination, in the
way described, of every pair of them, leads to an equation which is
linearly deducible from them. If the number of this so-called
_complete system_ is n, the equations give df/dx1 = 0 ... df/dxn = 0,
leading to the nugatory result f = a constant. Suppose, then, the
number of this system to be r < n; suppose, further, that from the
matrix of the coefficients a determinant of r rows and columns not
vanishing identically is that formed by the coefficients of the
differential coefficients of f in regard to x1 ... x_r; also that the
coefficients are all developable about the values x1 = x1^0, ... xn=
xn^0, and that for these values the determinant just spoken of is not
zero. Then the main theorem is that the complete system of r
equations, and therefore the originally given set of k equations,
have in common n - r solutions, say [omega]r+1, ... [omega]n, which
reduce respectively to x_r+1, ... x_n when in them for x1, ... x_r are
respectively put x1^0, ... x_r^0; so that also the equations have in
common a solution reducing when x1 = x1^0, ... x_r = x_r^0 to an
arbitrary function [psi](x_r+1, ... x_n) which is developable about
x_r+1^0, ... x_n^0, namely, this common solution is [psi]([omega]_r+1,
... [omega]_n). It is seen at once that this result is a
generalization of the theorem for r = 1, and its proof is conveniently
given by induction from that case. It can be verified without
difficulty (1) that if from the r equations of the complete system we
form r independent linear aggregates, with coefficients not
necessarily constants, the new system is also a complete system; (2)
that if in place of the independent variables x1, ... xn we introduce
any other variables which are independent functions of the former, the
new equations also form a complete system. It is convenient, then,
from the complete system of r equations to form r new equations by
solving separately for df/dx1, ..., df/dx_r; suppose the general
equation of the new system to be
Q_[sigma]f = df/dx_[sigma] + c_[sigma],r+1 df/dx_r+1 + ... + c_[sigma]n df/dx_n = 0 ([sigma] = 1, ... r).
Then it is easily obvious that the equation Q_[rho]Q_[sigma]f -
Q_[sigma]Q_[rho]f = 0 contains only the differential coefficients of f
in regard to x_r+1 ... xn; as it is at most a linear function of Q1f,
... Qrf, it must be identically zero. So reduced the system is called
a Jacobian system. Of this system Q1f=0 has n - 1 principal solutions
reducing respectively to x2, ... xn when
x1 = x1^0,
and its form shows that of these the first r - 1 are exactly x2 ...
xr. Let these n - 1 functions together with x1 be introduced as n new
independent variables in all the r equations. Since the first equation
is satisfied by n - 1 of the new independent variables, it will
contain no differential coefficients in regard to them, and will
reduce therefore simply to df/dx1 = 0, expressing that any common
solution of the r equations is a function only of the n - 1 remaining
variables. Thereby the investigation of the common solutions is
reduced to the same problem for r - 1 equations in n - 1 variables.
Proceeding thus, we reach at length one equation in n - r + 1
variables, from which, by retracing the analysis, the proposition
stated is seen to follow.
System of total differential equations.
The analogy with the case of one equation is, however, still closer.
With the coefficients c_[sigma]j, of the equations Q_[sigma]f = 0 in
transposed array ([sigma] = 1, ... r, j = r + 1, ... n) we can put
down the (n - r) equations, dx_j = c1_j dx1 + ... + c_rj dx_r,
equivalent to the r(n - r) equations dx_j/dx_[sigma] = c_[sigma]r.
That consistent with them we may be able to regard x_r+1, ... x_n as
functions of x1, ... x_r, these being regarded as independent
variables, it is clearly necessary that when we differentiate
c_[sigma]j in regard to x_[rho] on this hypothesis the result should
be the same as when we differentiate c[rho]j, in regard to x[sigma] on
this hypothesis. The differential coefficient of a function f of x1,
... xn on this hypothesis, in regard to x_[rho]j is, however,
df/dx_[rho] + c_[rho],r+1 df/dx_r+1 + ... + c_[rho]n df/dx_n,
namely, is Q_[rho]f. Thus the consistence of the n - r total equations
requires the conditions Q_[rho]c_[sigma]j - Q_[sigma]c_[rho]j = 0,
which are, however, verified in virtue of Q[rho](Q[sigma][f]) -
Q_[sigma](Q_[rho]f) = 0. And it can in fact be easily verified that if
[omega]_r+1, ... [omega]_n be the principal solutions of the Jacobian
system, Q_[sigma]f = 0, reducing respectively to x_r+1, ... xn when x1
= x1^0, ... x_r = x_r^0, and the equations [omega]_r+1 = x_r+1^0, ...
[omega]_n = x_n^0 be solved for x_r+1, ... x_n to give x_j =
[psi]_j(x1, ... x_r, x_r+1^0, ... x_n^0), these values solve the total
equations and reduce respectively to x_r+1^0, ... x_n^0 when x1 = x1^0
... x_r = x_r^0. And the total equations have no other solutions with
these initial values. Conversely, the existence of these solutions of
the total equations can be deduced a priori and the theory of the
Jacobian system based upon them. The theory of such total equations,
in general, finds its natural place under the heading _Pfaffian
Expressions_, below.
Geometrical interpretation and solution.
Mayer's method of integration.
A practical method of reducing the solution of the r equations of a
Jacobian system to that of a single equation in n - r + 1 variables
may be explained in connexion with a geometrical interpretation which
will perhaps be clearer in a particular case, say n = 3, r = 2. There
is then only one total equation, say dz = adz + bdy; if we do not take
account of the condition of integrability, which is in this case da/dy
+ bda/dz = db/dx + adb/dz, this equation may be regarded as defining
through an arbitrary point (x0, y0, z0) of three-dimensioned space
(about which a, b are developable) a plane, namely, z - z0 = a0(x -
x0) + b0(y - y0), and therefore, through this arbitrary point [oo]^2
directions, namely, all those in the plane. If now there be a surface
z = [psi](x, y), satisfying dz = adz + bdy and passing through (x0,
y0, z0), this plane will touch the surface, and the operations of
passing along the surface from (x0, y0, z0) to
(x0 + dx0, y0, z0 + dz0)
and then to (x0 + dx0, y0 + dy0, Z0 + d^1z0), ought to lead to the same
value of d^1z0 as do the operations of passing along the surface from
(x0, y0, z0) to (x0, y0 + dy0, z0 + [delta]z0), and then to
(x_ + dx_ , y_ + dy_ , Z_ + [delta]^1z_ ),
0 0 0 0 0 0
namely, [delta]^1z0 ought to be equal to d^1z0. But we find
d^1z0 = a0dx0 + b(x0 + dx0 , y0, z0 + a0dx0)dy0 =
/db db \
a0dx0 + b0dy0 + dx0dy0( --- + a0--- ),
\dx0 dz0/
and so at once reach the condition of integrability. If now we put x
= x0 + t, y = y0 + mt, and regard m as constant, we shall in fact be
considering the section of the surface by a fixed plane y - y0 = m(x -
x0); along this section dz = dt(a + bm); if we then integrate the
equation dx/dt = a + bm, where a, b are expressed as functions of m
and t, with m kept constant, finding the solution which reduces to z0
for t = 0, and in the result again replace m by (y - y0)/(x - x0), we
shall have the surface in question. In the general case the equations
dx_j - c_1j dx1 + ... c_rj dx_r
similarly determine through an arbitrary point x1^0, ... xn^0 a planar
manifold of r dimensions in space of n dimensions, and when the
conditions of integrability are satisfied, every direction in this
manifold through this point is tangent to the manifold of r
dimensions, expressed by [omega]_r+1 = x_r+1^0, ... [omega]_n = x_n^0,
which satisfies the equations and passes through this point. If we put
x1 = x1^0 = t, x2 = x2^0 = m2t, ... xr = xr^0 = mrt, and regard m2,
... mr as fixed, the (n-r) total equations take the form dx_j/dt =
c_1j + m2c_2j + ... + m_rc_rj, and their integration is equivalent to
that of the single partial equation
n
df/dt + [Sigma](c_1j + m2c_2j + ... + m_rc_rj)df/dx_j = 0
j=r+1
in the n - r + 1 variables t, xr+1, ... xn. Determining the solutions
[Omega]_r+1, ... [Omega]_n which reduce to respectively x_r+1, ... x_n
when t = 0, and substituting t = x1 - x1^0, m2 = (x2 - x2^0)/(x1 -
x1^0), ... mr = (xr - xr^0)/(x1 - x1^0), we obtain the solutions of
the original system of partial equations previously denoted by
[omega]_r+1, ... [omega]_n. It is to be remarked, however, that the
presence of the fixed parameters m2, ... mr in the single integration
may frequently render it more difficult than if they were assigned
numerical quantities.
Pfaffian Expressions.