This work has been considerably enlarged since the publication of the Fourth Edition. Not only does the Volume contain some 45 pages more than its predecessor, but further space has been gained by the use of smaller type for merely illustrative matter. The most important addition in the body of the work is the very far-reaching theorem which I have given as Prop. LVII. Which enables us to write down at sight the mean value of such functions as a3, &c., when a, B, 7, are the parts into which a given magnitude is divided at random. I first published this theorem in a pamphlet in the year 1898.