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THE Author of this Essay has been forewarned that the adoption of the subscript differential notation, which forty years ago was in some degree of favour but afterwards laid aside, will prove a serious hinderance to the acceptableness of the present Work, as it will be new to the generality of mathematical students, and therefore somewhat repellent. He regrets the circumstance, and feels that an explanation and apology are needed; and he finds both in the fact that the notation alluded to has been forced upon him by the necessities of the case; and, moreover, he ventures to think that the reader after going through the Essay will agree with him in this, and be willing on this ground to excuse such an important deviation from established mathematical usage.
The Author denominates his Work an Essay, because as a TREATISE it would be very incomplete, there being many portions of the general subject not alluded to in these pages. For information on those portions he must refer the reader to Boole and Gregory and Carmichael, with whose admirable and valuable Works he has been careful to interfere as little as possible. In fact, he has consulted them only for such illustrative examples as fell within the scope of his Essay. He is conscious of having apparently intermeddled with the integration of equations of one independent variable; but this has arisen out of the peculiarities of the Author's system, by which every equation, whatever be the number of its independent variables, is reduced to an equivalent equation of only one independent variable. It was therefore impossible to keep the two branches of differential equations apart.
The system of integration here proposed occurred to the mind of the Author a few years ago, but his professional engagements did not then leave him leisure to follow the general idea into its details. And, as his object is merely to render his Method thoroughly intelligible, and not the exhibition of integrals, he has for brevity's sake not always proceeded to the last steps of an integration when he conceived that the method had been made intelligible.
The Author fears a cursory glance at the pages of the Work will have a prejudicial effect; for he is aware that some of them exhibit a formidable and deterrent array of novel symbols; he therefore begs to assure the reader that the various steps of the investigations are all obedient to one general principle, and though in some degree novel, are not really difficult, but on the contrary easy when the eye has become accustomed to the novelties of the notation. And, moreover, he entertains a hope that the results of integrations (many of which are far more general than they were in the shape in which they have appeared in former Treatises) will repay the reader for any extra trouble he may find in pushing through what may at first appear to