## Abstract

The preceding papers of this series have examined the properties of an optical calculus which represented each of the separate elements of an optical system by means of a single matrix **M**. This paper is concerned with the properties of matrices, denoted by **N**, which refer not to the complete element, but only to a given infinitesimal path length within the element.

If **M** is the matrix of the optical element up to the point *z*, where *z* is measured along the light path, then the **N**-matrix at the point *z* is defined by

Thus one may write symbolically,

and

A general introduction is contained in Part I. The definition and general properties of the **N**-matrices are treated in Part II. Part III contains a detailed discussion of the important special case in which the optical medium is homogeneous, so that **N** is independent of *z*; Part III contains in Eq. (3.26) the explicit relation which corresponds to the symbolic relation (C). Part IV describes a systematic method, based on the **N**-matrices, by which the optical properties of the system at each point may be described uniquely and quantitatively as a combination of a certain amount of linear birefringence, a certain amount of circular dichroism, etc.; the method of resolution is indicated in Table I. Part V treats the properties of the inhomogeneous crystal which is obtained by twisting a homogeneous crystal about an axis parallel to the light path.

© 1948 Optical Society of America

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