9. The general problem of the mathematical theory of invariants may now be restated as follows: Given any group of transformations, and any set of configurations (mathematical objects) which are converted into one another by these transformations, to find the properties of the configurations which remain unaltered.
There are thus as many types of invariant theories as there are types of groups and types of configurations. The distinction according to groups is the more fundamental.
10. The relation between the concepts in variant and group is thus very intimate. Suppose that instead of assigning transforma tions and seeking invariants, the question is inverted: Given a property or a function, find the transformations which leave it invariant.
Thus, if space is to be deformed so as to leave the distance between every pair of points unaltered, the deformation must be simply a displacement. The totality of displacements forms a group; for if one displacement is fol lowed by a second, the effect is the same as that due to a single displacement. In general the totality of transformations defined by one or more invariants is a group. Many groups are
defined in this way.
11. In connection with any group this ques tion may be asked: Given two configurations, is it possible to convert the one into the other by a transformation belonging to the assigned group? If this is the case, the configurations are said to be equivalent with respect to the group, Thus, two figures are equivalent with respect to the displacement group when they are con gruent; they are equivalent with respect to the similitude group when they are similar. The study of equivalence with respect to the pro jective group is the main object of projective geometry; its systematic analytic treatment depends upon the theory of algebraic forms considered below.
The importance of the notion of equivalence depends upon the fact that equivalent configu rations necessarily have the same invariant func tions and properties. Thus in studying the pro jective properties of (proper) conics, It is suffi cient to consider the case of a circle. The cir cle may thus be taken as the type or canonical form of the class of conics.