After a brief review of the nature of the calculus, a formulation is provided for the application of the calculus to a wider class of optical systems than those postulated in the earlier papers of this series. The application is found, however, to be subject to the drastic limitation that the system must not depolarize the light passing through it. Since homogeneous crystalline plates are almost completely free of depolarization, this limitation does not exist in the previously contemplated application to piles of crystalline plates. A more powerful calculus, whose formulation is due to Soleillet and whose development is due to Perrin and Mueller, is described, and is found to be free of this limitation. Its general properties are compared with those of the calculus discussed in this series of papers.
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