In a former memoir on this subject, it was my endeavour to exhibit the principles of the science of General Differentiation in a simple, at the same time in a general, point of view. I endeavoured to deduce, from one general formula, results easy of application in all instances; and thus to exhibit the unity of the different parts of the science, and the completeness of its fundamental formulæ, shewing at the same time the facility of their adaptation to particular and varied cases. With the exception of certain expansions by means of a theorem analogous to the series of Taylor, I gave no application of the principles to problems of any kind. It is my intention in the present memoir to supply this branch of the subject, without which, indeed, however interesting may be the details, as a portion of pure analysis, they will offer little to interest any but those who attach themselves to the study of analytical combination. We hope, by the exhibition of a few simple mechanical problems, solved by this process, to give to our subject an interest in the eyes of all, derived not from its intrinsic beauty, but from its use as a medium of demonstration. It is well known that considerable difficulty hangs over several very simple inverse mechanical problems; from the generality of their statement, a direct solution is sometimes impossible by the ordinary methods. We shall shew that by our process such solutions are attainable with the greatest readiness. By this means we hope to give a value to our subject as a branch of knowledge, independent of that value which it must possess from its curious and elegant structure.