INVARIANTS AND COVARIANTS.
I. These terms were introduced and are still ordinarily employed, in connection with a special mathematical theory, namely, the theory of the linear transformation of algebraic forms de veloped by Cayley and Sylvester dining the middle third of the 19th century. The central idea, however, is a very general one, which has been applied in recent years to almost all branches of mathematics. It deserves, in fact, to be ranked with such fundamental concepts as function and group. We therefore .divide our sketch into three parts as follows: (1) The' general concept of invariant; (2) the theory of algebraic forms, or invariants in the narrow sense; (3) other invariant theories.
2. The suggestion for the formation of the concept comes from the familiar observation, at the bottom of all science and philosophy, that, while the world about us is in a continual state of change, there are yet certain aspects or properties which are unaltered. To find the permanent in the changing is the most general statement of the problem of invariants. Ab stractly, the idea may be explained more de finitely as follows: Consider a set of objects or elements 0 of any conceivable kind, finite or infinite in number; and a set of operations or transformations T, each of which interchanges the objects in a definite manner. Then a prop erty of an object 0 is said to be invariant, provided it holds for all the objects obtained from the given 0 by the transformations T. Similarly, any relation between a number of 0's which holds for the transformed O's is said to be an invariant relation, that is, an invariant relation of the given objects with respect to the given transformation.
The idea of covariant involves nothing essen tially new. An object 15 is said to be a covari ant of a given number of objects 0., Oa etc.. provided 0 is invariantly related to 01, 0,,.... In this case, if any one of the transformations T converts O into 0', Oi into 0.', 0, into 0,', etc., then the relations connecting O' with 0.. 0., etc., are the same as those connecting 0 with 0., 0., etc.
3. The idea is best illustrated by examples from geometry. Consider .a number of points P,, P., ..., connected with a solid body. When the body is displaced, its points take new positions, P., .... Many such positions are possible, since the displacement may be made in an endless number of ways. But in
every base, of course, the distance between P.' and P.' is the same as that between P. and P.. That is, distances between points are invariant with respect to rigid displacement.
Suppose next that the solid carrying the points is not only displaced but is magnified' (or diminished) according to any scale. (We. may, for example, picture such a change as produced by subjecting the homogeneous solid to a higher or lower temperature). The solid is then converted into one of different size but of the same shape, that is, a similar. solid. Dis tances are changed in the same ratio. Hence P,'P.'/P,'Pl'P.P./P2P.. That is, the ratio of any two distances is invariant with respect to 'similitude transformations.
4, In both examples, points on a straight line are converted into points on a straight line. Collinearity is then a relation which is in variant with respect to displacements and simili tude transformations. A more general type of transformation for which this is true is the homographic or projective transformation. We considm-, for simplicity,. only the case of figures drawn in a plane M. From a fixed point (termed the centre of projection) outside of M draw lines to the various points of M until they intersect a second plane M'. Thus, every point P in M is associated with a definite point P' in The operation of passing from a figure in M to the corresponding figure in M' is termed projection. Concretely, we may think of the centre of projection as a source of light and the figure in Al' as the shadow of that in M.* If we consider three points Ph P., P. on a straight line in M, they are converted, by projection, into points P.' on a straight line in M'. But in general the distances and, also the ratios of distances will differ. In fact, three points have no invariant, since they may be converted into three points at arbitrarily assigned distances by a suitable projection. If, however, we take four points (on a straight line) it may be shown that, for any projection, PIP/ 77r7 • In each member of this equation we have a combination of the distances between four points which is termed their cross ratio (an harmonic ratio). Hence the cross ratio of four, collinear points is invariant with respect to projective transformation.