The first-order optical nature of an optical system (including an eye) is completely characterized by a 55 ×  matrix called the ray transference.  It is known  that the image of a ray transference by the matrix logarithm function is an augmented Hamiltonian matrix.  It turns out that there are other ways of transforming transferences into augmented Hamiltonian matrices.  They include Cayley transforms and modified Cayley transforms.  This paper will describe these transforms with a view to finding the most suitable one for quantitative analyses of eyes and other systems in augmented Hamiltonian spaces.  In particular we look at the calculation of average systems.