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 + d’u du and + = 0. dx* dydz 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 vari able only. Quasi-constants. 1. If-t, x, y, z ... be n independent variables, and u the dependent variable, it is understood that du, dzu, dyu, dou, 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 do, then t, y, 2, ... 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 de, we shall denominate x, y, z, ... and any functions of them quasi-constants : the use of d., will imply that t, y, % ... and their functions are quasi-constants; and so on. 2. If s4 be the symbol of integration corresponding to de St 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, , ... 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. = Change of the independent variables. 3. If from the (n) independent variables t, X, Y, Z, ... We wish to pass to a new set 7, g, n, & ..., formulæ for this purpose are ready to hand in the Differential Calculus. They are of the following type: (d-u) = dyu. (dut) + dxu.(dx)+dyu.(dxy) + ......(1). This is only one of the (n) requisite equations, and the other (n-1) equations may be obtained from this by writing successively in this , n, % ... for T. But it is possible, whenever the forms of , n, % ... are at our disposal, to assign them such values in terms of t, X, Y, Z, ... as shall render the above equation (1) sufficient by itself for our purpose; so that by a proper choice of values for g, n, , ... any partial differential 1, % 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 assigning proper forms for the quasi-constants g, n, t, ... in terms of t, x, y, 2, ... the original independent variables will t2 be best learned from the examples which will be given. Distinctive Symbols of operation for different sets of inde pendent variables. 4. In the formula (1) certain terms are enclosed in brackets; this is done to indicate that before the differential operations indicated in them are performed, t,a, y... must be expressed in terms of the new variables; or, that the operations are to be performed on the supposition that u is so expressed. In brief, the distinction of brackets and no brackets in equation (1) refers to the fact that there are operations to be performed in that equation in reference to two sets of independent variables. Had we to write down an equation in which differential and integral operations would have to be performed in reference to three or more sets of independent variables, we should have to employ two or more kinds of distinctive brackets to indicate that fact. Now it is proposed in this Essay to indicate this fact by employing distinctive differential and integral symbols of operation, and so to abolish the distinction by brackets. Thus, when t, g, n, %... is the set T of independent variables referred to in a differential or integral operation, we shall use D and S as the symbols of operation; if T', ', n', $, ... be the set of independent variables, we shall indicate that fact by using A and E as the symbols of operation; and so on for any number of sets. Under this system equation (1) of Art. 3 will be written without brackets thus, Dqu= d.u.Dat+dzu.Dqx+d,u.Dry + ......... (1). 5. In speaking about different sets of independent variables, it is to be noticed that it is not necessary that every individual member of one set should be different from every |