^{page no 1 note *} The number δ_αα remains unaltered only when the orientation of the coordinate axes is preserved, one change of sense in this transforming δ_αα to δ_αα The invariant depends, therefore, on the two triangles and the positive sense of rotation adopted in the plane.

^{page no 2 note *} We express the fact that a number z, which is in general complex, has in a particular instance a purely real value, without specifying that value, by writing z = [r]. Similarly, the fact that z is purely imaginary is denoted by writing z =[pi].

^{page no 2 note †} The pairs of homologous vertices are, besides, the same in the two correspondences in which the triangles are orthologic and metaparallel.

^{page no 4 note *} See Agronomof, , Revista matemática hispano-americana, 1927, pp. 299–304.Google Scholar

^{page no 5 note *} |φ| = 1 implies the existence of one point common to the three circles, but as a consequence of this, the circles meet in a second point. For, if P is the first point common to the three circles, there exists another point Q through which the circles pass, since the equations

are certainly compatible. P and Q are, besides, the two points in the plane whose distances from the vertices A _l, B ₁, C ₁ are proportional to three given lengths.