Three Dimensional Chains and the Classification of the Resulting Collineations in Space: Thesis (Classic Reprint) Hardback -

If, however, the coefficients in the transformation are restricted to real numbers and if two such transformations are regarded as equivalents only if one can be transformed into the other by a transformation with real coefficients the number of types is increased. Thus in the one-dimensional case we have three well known types usually designated by the terms+, hyperbolic, elliptic, and, .parabolic. Here, furthermore, arises the important additional problem as to the conditions under which a collineation with complex coefficients can be transformed into one with real coefficients. In the one - dimensional case again these confiitions are well known: it is necessarv and sufficient that the projective transformation on the complex variable leave a circle in the Argand plane of the complex variable invariant.

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