Excerpt from The Doctrine of Germs: Or the Integration of Certain Partial Differential Equations Which Occur in Mathematical Physics

In Chap. III. Is introduced a theory of Symbolical Equivalences. The subject is regarded from a point of view which may be considered as in some degree new. The exigencies of this Essay did not seem likely to require the complete development of this Theory; and in consequence of this only so much detail is given as was likely to he wanted in subsequent chapters. The Theory is capable of throwing light on several troublesome known paradoxes which have often been a source of perplexity to the Mathematical Student.

As the author was induced to undertake the development of the Doctrine of Germs by a desire to accomplish the complete integration of Laplace's Equation, and the consequent discovery of the general form of Laplace's Functions, he has deemed it to be of some possible advantage to obtain and to exhibit those results in several forms and aspects; hoping also that by this diversity the reader would be the more disposed to accept results which so confirm one another.

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