In a previous paper, it was pointed out that the characteristic problem of the third branch of the higher calculus, is to discover the relation between the primary variable and its function, when the relation subsisting between the function and its derivative is known. The present paper treats of the solution of the simplest case of this general problem, that in which the function is equal or proportional to its derivative.

The proposition in hand is naturally divided into cases, according to the order of derivation: The first two of these can, by well-known artifices, be brought under the dominion of the integral calculus, and their relations can therefore present nothing new. But for the sake of the continuity of the treatment, and of certain relationships which otherwise could not have been so well explained they have been discussed in the paper.