VIII. On a general method in analysis

Abstract

Much attention has of late been paid to a method in analysis known as the calculus of operations, or as the method of the separation of symbols. Mr. Gregory, in his Examples of the Differential and Integral Calculus, and in various papers published in the Cambridge Mathematical Journal, vols. i. and ii., has both clearly stated the principles on which the method is founded, and shown its utility by many ingenious and valuable applications. The names of M. Servois (Annales des Mathé-matiques, vol. v. p. 93), Mr. R. Murphy (Philosophical Transactions for 1837), Professor De Morgan, &c., should also be noticed in connection with the history of this branch of analysis. As I shall assume for granted the principles of the method, and shall have occasion to refer to various theorems established by their aid, it may be proper to make some general remarks on the subject by way of introduction. Mr. Gregory lays down the fundamental principle of the method in these words: “There are a number of theorems in ordinary algebra, which, though apparently proved to be true only for symbols representing numbers, admit of a much more ex­tended application. Such theorems depend only on the laws of combination to which the symbols are subject, and are therefore true for all symbols, whatever their nature may be, which are subject to the same laws of combination.” The laws of combination which have hitherto been recognised are the following, π and ρ being symbols of operation, u and v subjects.