= 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 ord