Observations on the analogy which subsists between the calculus of functions and other branches of analysis

At the commencement of this paper the author states the advantages which may be derived from the employment of analogical reasoning in mathematics, and recommends it as a very useful guide to new discoveries: he then proceeds to point out the striking resemblance which subsists between several parts of common algebra and the integral calculus, and similar parts of the calculus of functions. Mr. Babbage then notices certain fractions which, by peculiar relations among the functions of which they consist, become evanescent. The true values of these fractions are ascertained, and they are applied to the solution of a class of functional equations which the author had solved in a former paper, from which the following result is obtained :—“ Whenever the mode of solution there adopted seems to fail, the failure isapparent only, and the general solution may always be deduced from it.”