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 à 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 him a forest of operative symbols. Many equations will be found in this Essay integrated with ease in finite terms, which, as far as the Author is aware, were never integrated in finite terms before. In the last Chapter, for instance, the following highly important equations will be found integrated for the first time in finite terms; = du du + + cu=0); dx2 dy" d2u du du and + + = 0, dx dy dz* Three more important partial differential equations could hardly be instanced, and a finite integral of each one of them is obtained in this Essay without any difficulty. The Author can scarcely hope that in a work so novel in its methods as this is, and written as this has been under considerable pressure of other engagements, some clerical errors will not have escaped his notice; but he hopes that they will not prove to be many, and that none of them will involve errors of principle. If any are found, he will be thankful if the courteous reader will send notice of them addressed to him at the Publishers. SHEFFIELD, October 2, 1871, AN ESSAY. CHAPTER I. GENERAL PRINCIPLES OF THE ESSAY. It is known that the integration of differential equations is oftentimes much facilitated by a change of one or more of the independent variables, and sometimes also by changing the dependent variable. The process is however often tedious, and it is frequently far from obvious what ought to be chosen as the new variables, even when it is seen that a change should be made. The system of integration to be proposed in this Essay is founded on a simple principle of choosing new variables; and the change of variables is effected with great ease. And the most important feature of the system is, that whatever be the number of original independent variables, the work of integration is at once reduced to the use of one independent variable only Quasi-constants. 1. If t, x, y, 2 ... be n independent variables, and u the % dependent variable, it is understood that du, deu, d,u, d,u, ... are partial differential coefficients; by which is meant that in finding the value of diu, i.e. in operating on u with the symbol de, we are to consider x, y, z, ... as constants : in operating on u with ds, then t, y, a, ... are to be considered as constants; and so on. It is here proposed to express this briefly by the use of the word quasi-constant. Thus in reference to the use of the symbol dt, we shall denominate x, y, z, and any functions of them quasi-constants : the use of d, will imply that t, y, z ... and their functions are quasi-constants; and so on. 2. If s be the symbol of integration corresponding to de the symbol of differentiation, then we know that after the operation Se we must add to complete the integral an arbitrary function of x, y, z, ... its quasi-constants: and this principle , is general and may be thus stated,-after an operation of integration we are to add to the integral an arbitrary function of the corresponding quasi-constants; and all the variables which were considered constants in differentiation are to be considered quasi-constants in integration. we Change of the independent variables. 3. If from the (n) independent variables t, x, y, 2, ....We wish to pass to a new set t, f, n, & ..., formulæ for this purpose are ready to hand in the Differential Calculus. They are of the following type: (d-u) = deu. (dzt) +dqu. (d-x) +du. (day) + ...... (1). This is only one of the (12) requisite equations, and the other (n-1) equations may be obtained from this by writing successively in this g, n, 5 ... for T. But it is possible, whenever the forms of ť, n, S... are at our disposal, to assign them g súch values in terms of t, X, Y, Z, ... as shall render the aboveequation (1) sufficient by itself for our purpose; so that by a proper choice of values for &, n, %, ... any partial differential equation of n variables may be reduced to an equivalent equation of one variable and (n − 1) quasi-constants. This is the principle on which this Essay is founded. The method of |