The subject of the following paper is the first proposition in the Calculus of Primaries. The business of that 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 simplest relationship between two variable quantities is proportionality when they are heterogeneous, or equality, when they are of one kind; and the case of proportionality can always, by a change in the unit of measure, be brought to an equality of the representative numbers; so that our first proposition becomes this: “To investigate the nature of those functions which reappear among their own derivatives.” Since this reappearance must necessarily be periodical, I shall use the name Functions with Recurring Derivatives.