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< of any order. Moreover, the group allowed by the
equation may quite well consist of extended contact transformations.
An important application is to the case where the differential
equation is the resolvent equation defining the group of
transformations or rationality group of another differential equation
(see below); in particular, when the rationality group of an ordinary
linear differential equation is integrable, the equation can be solved
by quadratures.
Consideration of function theories of differential equations.
Following the practical and provisional division of theories of
differential equations, to which we alluded at starting, into
transformation theories and function theories, we pass now to give some
account of the latter. These are both a necessary logical complement of
the former, and the only remaining resource when the expedients of the
former have been exhausted. While in the former investigations we have
dealt only with values of the independent variables about which the
functions are developable, the leading idea now becomes, as was long ago
remarked by G. Green, the consideration of the neighbourhood of the
values of the variables for which this developable character ceases.
Beginning, as before, with existence theorems applicable for ordinary
values of the variables, we are to consider the cases of failure of such
theorems.
A general existence theorem.
When in a given set of differential equations the number of equations is
greater than the number of dependent variables, the equations cannot be
expected to have common solutions unless certain conditions of
compatibility, obtainable by equating different forms of the same
differential coefficients deducible from the equations, are satisfied.
We have had examples in systems of linear equations, and in the case of
a set of equations p1 = [phi]1, ..., pr = [phi]r. For the case when the
number of equations is the same as that of dependent variables, the
following is a general theorem which should be referred to: Let there be
r equations in r dependent variables z1, ... zr and n independent
variables x1, ... xn; let the differential coefficient of z[sigma] of
highest order which enters be of order h[sigma], and suppose d^h_[sigma]
z_[sigma]/dx1^h_[sigma] to enter, so that the equations can be written
d^h_[sigma] z_[sigma]/dx1^h_[sigma] = [Phi]_[sigma], where in the
general d]]>
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