### On the Product of Two Quadro-Quadric Space-Transformations

American Journal of Mathematics, Vol. 35, No. 2 (April 1913), 183-188

Introduction

Although the analogy of the plane theory calls attention to the composition and decomposition of space-transformations, yet the subject has received very slight attention except from Herr S. Kantor, who has dealt fully with the general case of composition of quadro-quadric transformations. As so often in this theory, the more specialized cases are in some ways more interesting, and worth considering in detail.

The object of this paper is to show that the transformation compounded of two quadro-quadric transformations belongs to one of the following types:

- A quarto-quartic, whose fundamental system in either space consists of a double and a simple point, a nodal conic and a simple quartic.
- A cubo-quartic, with, in the first space, a double and a simple point, a simple cubic and conic.
- A cubo-cubic, with a double point, a simple quartic and conic.
- A cubo-cubic, with a simple cubic, conic and straight line.
- The general cubo-cubic with a nodal line.
- The general quadro-quartic.
- The general quadro-cubic.
- The general quadro-quadric.
- The general linear transformation, including identity.