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, z... be n independent variables, and u the dependent variable, it is understood that du, du, du, du, ... are partial differential coefficients; by which is meant that in finding the value of du, i.e. in operating on u with the symbol d, we are to consider x, y, z, ... as constants: in operating on u with d, then t, y, z,... 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 d, 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 d the symbol of differentiation, then we know that after the operations, 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.

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, §, n, ....., formulæ for this purpose are ready to hand in the Differential Calculus. They are of the following type:

(d,u) = d ̧u. (d ̧t)+d ̧u. (d ̧x) +du. (d,y) +

......

..(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, S, ... 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 aboveequation (1) sufficient by itself for our purpose; so that by a proper choice of values for E, 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

§,

assigning proper forms for the quasi-constants E, n,,... in terms of t, x, y, z,... the original independent variables will be best learned from the examples which will be given.

Distinctive Symbols of operation for different sets of independent 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,ix, 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 7, §, n, ... is the set 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,—

Du du. D,t+du.D,x+du. Dy +......... (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

individual member of another set. If some, even if one, of one set be different from all of another set this is sufficient to establish them as different sets, and to entitle them to the use of distinctive differential and integral symbols of operation.

6. In fixing the values of a new set of independent variables in terms of the original set, it is of importance to take care that each member of the new set be really an independent variable; i.e. that it be such as cannot be expressed in terms of the other members of its own set. Unless this condition of independence be observed, we shall be working under an erroneous hypothesis.

7. Under the authority of Art. 5 we are at liberty to assume, whenever it can be done usefully, that one of the variables of each new set shall be identical with one of the variables of the original set,-for example, we may assume T=t, in which case equation (1) of Art. 4 becomes

Du=du+du.D1x+du.D1y+.......

(1).

But when we use this equation, which will always be known from the occurrence of the symbol D, or St, we must remember that there remain only (n-1) new independent variables (the quasi-constants) of the set to be determined, one of them (t) being already fixed upon in using this equation.

8. In determining what shall be the forms of the constituent members of a new set of variables, we are at liberty to introduce n arbitrary hypotheses if we use equation (1) of Art. 4; but only (n-1) arbitrary hypotheses if we employ equation (1) of the last Article. The arbitrariness of these. hypotheses is however overruled by the necessity that the resulting members of a set of variables determined from them must be independent (Art. 6).

(5)

CHAPTER II.

DETERMINATION OF QUASI-CONSTANTS.

FIRST ORDER.

EQUATIONS OF THE

Ex. 1. To integrate du + ad ̧u = bu.

Let us change the independent variables to t, §. Then the formula of Art. 7 becomes for this case, which is one of two independent variables,

We are at liberty to make one hypothesis (Art. 8) for the determination of the value of §; let us assume Doc = a. We make this choice, because by this assumption the right-hand member of the formula (1) will be rendered identical with the left-hand member of the proposed equation; so that we shall have Du = bu;

.. Dxa, and Du=bu,

and these two equations, taken simultaneously, are together exactly equivalent to the proposed equation. Integrating them we have (see Art. 2)

x = at +, and e ̄u = F(§).

The former gives the value of §; and this value being written