en account of only in the bibliography.
It is believed that on the whole the article will be more useful to the
reader than if explanations of method had been further curtailed to
include more facts.

When we speak of a function without qualification, it is to be
understood that in the immediate neighbourhood of a particular set x0,
y0, ... of values of the independent variables x, y, ... of the
function, at whatever point of the range of values for x, y, ... under
consideration x0, y0, ... may be chosen, the function can be expressed
as a series of positive integral powers of the differences x - x0, y
-y0, ..., convergent when these are sufficiently small (see FUNCTION:
Functions of Complex Variables). Without this condition, which we
express by saying that the function is developable about x0, y0, ...,
many results provisionally stated in the transformation theories would
be unmeaning or incorrect. If, then, we have a set of k functions, f1
... fk of n independent variables x1 ... xn, we say that they are
independent when n >= k and not every determinant of k rows and columns
vanishes of the matrix of k rows and n columns whose r-th row has the
constituents dfr/dx1, ... dfr/dxn; the justification being in the
theorem, which we assume, that if the determinant involving, for
instance, the first k columns be not zero for x1 = x1^0 ... xn = xn^0,
and the functions be developable about this point, then from the
equations f1 = c1, ... fk = ck we can express x1, ... xk by convergent
power series in the differences x_k+1 - x_k+1^0, ... x_n - x_n^0, and so
regard x1, ... xk as functions of the remaining variables. This we often
express by saying that the equations f1 = c1, ... fk = ck can be solved
for x1, ... xk. The explanation is given as a type of explanation often
understood in what follows.

Ordinary equations of the first order.

Single homogeneous partial equation of the first order.

Proof of the existence of integrals.

We may conveniently begin by stating the theorem: If each of the n
functions [phi]1, ... [phi]n of the (n + 1) variables x1, ... x_nt be
developable about the values x1^0, ... x_n^0t^0, the n differential
equations of the form dx1/dt = [phi]1(tx1, ... xn) are satisfied by
convergent power series

x_r = x_r^0 + (t - t^0 ) A_r1 + (t - t0 )^2A_r2 + ...

reducing respectively to x1^0, ... xn^0 when t = t^0; and the only
functions satisfying the equations and reducin