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many sources among these may be mentioned, Carr's Synopsis of the Calculus; Legendre's Works on Elliptic Integrals and Gamma Functions; Mindung's Tafeln der Integralen; Sohncke's ditto; Schlömilch's Compendium; and Meyer's Definite Integrals.

The proofs have been read by both Mr R. F. Muirhead and the author, and it is hoped that not very many errors have escaped their repeated examinations. But the work is constructed on lines that are so novel that it cannot be hoped that it now appears without the need of some further correction. Any communication to the author giving him information of such slips will be received with sincere gratitude.

Critics who are schoolmen of the pure orthodox mathematical faith may find it hard to reconcile the ideas that have with them become innate, with some of the methods, and possibly some of the phraseology, here adopted. We only ask them to remember that there is arising a rapidly increasing army of men eagerly engaged in the development of physical research and in the industrial applications of scientific results, with the occasional help of mathematical weapons, whose mental faculties have been wholly trained by continuous contact with the hard facts of sentient experience and who find great difficulty in giving faith to any doctrine which lays its basis outside the limits of their experiential knowledge.

5th December 1896.

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