(i) Let X be the totality of points x = (x1,..., xn)for which all coordinates are different from zero, and let G be the group of transformations x i =

cxi, c > 0. Then a maximal invariant under G is (sgn xn, x1/xn,..., xn−1/xn)

where sgn x is 1 or −1 as x is positive or negative.

(ii) Let X be the space of points x = (x1,..., xn) for which all coordinates are distinct, and let G be the group of all transformations x i = f (xi),i = 1,..., n, such that f is a 1 : 1 transformation of the real line onto itself with at most a finite number of discontinuities. Then G is transitive over X .

[(ii): Let x = (x1,..., xn) and x = (x 1,..., x n) be any two points of X . Let I1,..., In be a set of mutually exclusive open intervals which (together with their end points) cover the real line and such that x j ∈ Ij . Let I 1,..., I n be a corresponding set of intervals for x 1,..., x n. Then there exists a transformation f which maps each Ij continuously onto I j , maps x j into x j , and maps the set of n − 1 end points of I1,..., In onto the set of end points of I 1,..., I n.]