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(1.) Before entering on the investigation which forms the object of this communication, it will be necessary to recall to recollection some general properties of the differences of the powers of the natural numbers, or of the numbers comprised in the general expression ∆m 0n , which I have elsewhere demonstrated, as well as to establish certain preliminary theorems by the aid of those properties, which will be useful in the progress of the inquiry. I shall employ throughout the separation of the symbols of operation from those of quantity, as respects ∆ and 0, in the manner followed in my paper “ On the Development of Exponential Functions,” published in the Philosophical Transactions, vol. cvi. p. 25 (1816), and further extended in its appli­cation in my “ Collection of Examples of the Applications of the Calculus of Finite Differences,” appended to the translation of Lacroix’s Differential and Integral Cal­culus in 1820, to which paper and collection the reader is referred for the demon­stration of the fundamental properties in question. (2.) Denoting by F(x ) any series of powers ofx , such as F (x )=Ax a +Bx b +Cx c +&amp;c., and byf (x ) any other asf (x )=Px p +Qx q +Rx r +&amp;c.

XIX. On the algebraic expression of the number of partitions of which a given number is susceptible