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<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.
We have above considered the integration of an equation
dz = adz + bdy
on the hypothesis that the condition
da/dy + bda/dz = db/dz + adb/dz.
It is natural to inquire what relations among x, y, z, if any, are
implied by, or are consistent with, a differential relation adx + bdy
+ cdx = 0, when a, b, c are unrestricted functions of x, y, z. This
problem leads to the consideration of the so-called _Pfaffian
Expression_ adx + bdy + cdz. It can be shown (1) if each of the
quantities db/dz - dc/dy, dc/dx - da/dz, da/dy - db/dz, which we shall
denote respectively by u23, u31, u12, be identically zero, the
expression is the differential of a function of x, y, z, equal to dt
say; (2) that if the quantity au23 + bu31 + cu12 is identically zero,
the expression is of the form udt, i.e. it can be made a perfect
differential by multiplication by the factor 1/u; (3) that in general
the expression is of the form dt + u1dt1. Consider the matrix of four
rows and three columns, in which the elements of the first row are a,
b, c, and the elements of the (r+1)-th row, for r = 1, 2, 3, are the
quantities u_r1, u_r2, u_r3, where u11 = u22 = u33 = 0. Then it is
easily seen that the cases (1), (2), (3) above correspond respectively
to the cases when (1) every determinant of this matrix of two rows and
columns is zero, (2) every determinant of three rows and columns is
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