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< = 0. Without going into further details, it is then clear
enough that the resolvent equation, being irreducible and such that
any solution is expressible rationally, with p parameters, in terms of
the solution [omega], enables us to define a linear homogeneous group
of transformations of y1 ... yn depending on p parameters; and every
operation of this (continuous) group corresponds to a rational
transformation of the solution of the resolvent equation. This is the
group called the _rationality group_, or the _group of
transformations_ of the original homogeneous linear differential
equation.
The group must not be confounded with a subgroup of itself, the
_monodromy group_ of the equation, often called simply the group of
the equation, which is a set of transformations, not depending on
arbitrary variable parameters, arising for one particular fundamental
set of solutions of the linear equation (see GROUPS, THEORY OF).
The fundamental theorem in regard to the rationality group.
The importance of the rationality group consists in three
propositions. (1) Any rational function of y1, ... yn which is
unaltered in value by the transformations of the group can be written
in rational form. (2) If any rational function be changed in form,
becoming a rational function of y1, ... yn, a transformation of the
group applied to its new form will leave its value unaltered. (3) Any
homogeneous linear transformation leaving unaltered the value of every
rational function of y1, ... yn which has a rational value, belongs to
the group. It follows from these that any group of linear homogeneous
transformations having the properties (1) (2) is identical with the
group in question. It is clear that with these properties the group
must be of the greatest importance in attempting to discover what
functions of x must be regarded as rational in order that the values
of y1 ... yn may be expressed. And this is the problem of solving the
equation from another point of view.
LITERATURE.--([alpha]) _Formal or Transformation Theories for
Equations of the First Order_:--E. Goursat, _Lecons sur l'integration
des equations aux derivees partielles du premier ordre_ (Paris, 1891);
E. v. Weber, _Vorlesungen uber das Pfaff'sche Problem und die Theorie
der partiellen Differentialgleichungen erster Ordnung_ (Leipzig,
1900); S. Lie un]]>
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