_Different Orders of Differentiation._ In the logical classification of the differential calculus which has just been given, some may be inclined to suggest a serious omission, since I have not subdivided each of its four essential parts according to another general consideration, which seems at first view very important; namely, that of the higher or lower order of differentiation. But it is easy to understand that this distinction has no real influence in the differential calculus, inasmuch as it does not give rise to any new difficulty. If, indeed, the differential calculus was not rigorously complete, that is, if we did not know how to differentiate at will any function whatever, the differentiation to the second or higher order of each determinate function might engender special difficulties. But the perfect universality of the differential calculus plainly gives us the assurance of being able to differentiate, to any order whatever, all known functions whatever, the question reducing itself to a constantly repeated differentiation of the first order. This distinction, unimportant as it is for the differential calculus, acquires, however, a very great importance in the integral calculus, on account of the extreme imperfection of the latter. _Analytical Applications._ Finally, though this is not the place to consider the various applications of the differential calculus, yet an exception may be made for those which consist in the solution of questions which are purely analytical, which ought, indeed, to be logically treated in continuation of a system of differentiation, because of the evident homogeneity of the considerations involved. These questions may be reduced to three essential ones. Firstly, the _development into series_ of functions of one or more variables, or, more generally, the transformation of functions, which constitutes the most beautiful and the most important application of the differential calculus to general analysis, and which comprises, besides the fundamental series discovered by Taylor, the remarkable series discovered by Maclaurin, John Bernouilli, Lagrange, &c.: Secondly, the general _theory of maxima and minima_ values for any functions whatever, of one or more variables; one of the most interesting problems which analysis can present, however elementary it may now have become, and to the complete solution of which the differential calculus naturally applies: Thirdly, the general determination of the true value of functions which present themselves under an _indeterminate_ appearance for certain hypotheses made on the values of the corresponding variables; which is the least extensive and the least important of the three.