pendent
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