5. Let the figures considere' be all the ellipses of a plane. With respect to displace ment an ellipse has two invariants, the major and the minor axis. With respect to similitude transformation, there is one invariant, the ratio of the axes, or what is essentially the same, the eccentricity. Finally, in the projective theory there are no invariants, since one ellipse may be converted into any other (and even into any proper conic).
In connection we may illustrate the notion of a covariant. The centre of an ellipse is a covariant with respect to displacement and magnification, but not with respect to pro jection. For if the plane containing an ellipse E and its centre C is displaced or magnified, so that ellipse E is converted into another ellipse E' and the point C is converted into point C', then C' is necessarily the centre of E'; while under projection this is not the case. A similar result holds for the centre of gravity of any figure, plane or solid.
6 Another well-known type of transforma tion is that known as inversion. Take a fixed circle F with centre C and radius r, and sup pose that any point P of the plane is converted into the point P' situated on the line CP, so that The points P, P are then said to be inverse with respect to the circle F. By the inverse of a curve is meant the locus of the points inverse to the points of the curve. The collinear relation of points is no longer in variant, for a straight line (not passing through C) is converted into a circle. An arbitrary circle is converted into a circle, but the centre of the circle is not a covariant point. The most important property of the transformation is this: the angle at which any two curves inter sect is equal to the angle at which the inverse curves intersect. Angles are invariant with respect to geometric inversion.
7. We pass now to a few simple examples of the general definition in No. 2, in which the objects and transformations are analytic in stead of geometric.
Let the objects 0 be functions of .any num ber of variables, and let the operations T per formed on these functions be the permutation of the variables involved. A function written down at random, for example x' 2ys, changes its form when say x and y are interchanged.
There are exceptional functions, like and xy+yz-Fxz, which are not altered by inter changing the variables in any way, and are termed symmetric. The symmetric functions are invariant with respect to permutation of the variables.
In the differential calculus it is shown that the exponential function sag has the property of being its own derivative. The only functions which are invariant with respect to the process of differentiation are in fact those of the form as*, where a is a constant. It is obvious that if the first derivative is equal to the original function, all the higher derivatives will also be equal to the function.
The trigonometric functions have a period of ter or 360°. Such a function f(x) is un altered in value when x is replaced by It is obvious that the double application of the operation, that iS, the replacing of x by .1-4-4r, will also leave f (x) invariant. The periodic character thus involves the invariance of the function with respect to all the operations (x, x+2kar) (this denotes the replacing of x by x+Ver), where k is any integer. We note here that if one of these operations, say that of adding 2k'ir to the angle, is followed by another, say that of adding 2k"ar, the result is the same as the single operation of adding 2(k'+k")tr, which is a member of the set. The set of operations thus possesses the essential property of a group.
8. In general, if an invariant is found with respect to certain operations, Ts, Ts, ..., the invariance also holds for all combinations of these operations. Adding these combinations to the original operations, a set is finally ob tained with the group property; this is then termed the group generated by the given opera tions.
Thus, in connection with No. 6, since angles are unaltered by inversion with respect to any circle, it follows that they are unaltered by successive inversion with respect to a number of circles. The inversions themselves do not con stitute a group; but the totality of combina tions is the important group of circular trans formations which are expressed analytically by the linear transformations of a complex variable and of its conjugate.