The primordial coefficients of asymmetrical lenses

An easily calculable system of sixteen magnitudes is constructed for the representation of the properties of asymmetrical lenses by the addition of four lengths to the magnitudes previously used. The equations connecting this system and the coefficients of the eikonal and of the characteristic function, the equations for combining systems or moving their reference points, and the identities between the coefficients are expressed in matrix form. It is shown that the eleven variables present in a system of three separated astigmatic lenses only yield nine degrees of freedom, and such a system cannot, if the reference points are placed where the outer lenses meet the axis, represent the general system, which has ten degrees of freedom.