dv

dP

Hence since is the coefficient of in A'P, and

dx;

dp;

du

dxj

its coefficient in AP, its co efficient in the derivedequation (3) will be (Chap. xxv. Art. 5),

du d2v

du d2v dv d'u

dv d2u

+

\dx; dp¡dx; dp; dx¡dx; dx; dp¡dx; dp¡ dxdx;

whence the Proposition is established.

Applying this to the system (2) we see that any derived equation will be of the form

[F,F;] P | = 0.

But [FF]=0 by the conditions given; hence the condition (A‚— A‚A1) P =0, is identically satisfied.

The results of Chapter XXVI. being thus directly applicable to the system under consideration, we see that a common integral of the system (2) may be found by a series of alter

nate processes of integration and derivation. We begin by seeking an integral of the first partial differential equation. By a process of derivation, always possible, followed by the integration of a differential equation between two variables, we arrive at a common integral of the first two partial differential equations. Again, by a process of derivation followed by the solution of a differential equation we obtain a common integral of the first three partial differential equations. And so on, until a common integral of all is obtained.

7. Another solution of the above problem has recently been given. Beginning as in Jacobi's method by finding an integral of the first partial differential equation, a process of derivation agreeing in principle with Jacobi's, only more extended, may lead us without further integration to a point. at which the discovery of a common integral of the entire system will depend only upon the solution of a single differential equation of the first order susceptible of being made integrable by a factor. Failing this, it will enable us to convert the given system of partial differential equations into a new system possessing the same general character, but containing one equation less. Upon this the same process may be tried with a similar final alternative-and so on till the required integral is discovered. (On the Differential Equations of Dynamics. Philosophical Transactions, 1863).

CHAPTER XXVIII.

PARTIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER.

[THIS Chapter is a reconstruction on a larger scale of part of Chapter XV. At the end of the Chapter reference will be given to other writings of Professor Boole on the subject here discussed.]

1. The general form of a partial differential equation of the second order is

F(x, y, z, p, q, r, s, t) = 0............(1),

where

It is only in particular cases that the equation admits of integration, and the most important is that in which the differential coefficients of the second order present themselves only in the first degree; the equation thus assuming the form

in which R, S, T, and V are functions of x, y, z, p and 4.

The most important part of the theory of the solution of this equation is due to Monge, and was extended by Ampère to the more general equation

Rr + Ss + Tt + U (s2 — rt) = V ........ ..(3).

This equation, together with the particular equation of Monge, and the equation.

Rr + Ss + Tt + U (s2 — rt) = 0,

both which though falling under Ampère's general form possess peculiarities demanding special notice, I propose to consider in this Chapter. I shall in conclusion make some observations on the theory of partial differential equations of the second order with more than two independent variables.

Monge's method, and Ampère's in so far as it is an extension of Monge's, consists in a certain procedure for discovering either one or two first integrals of the form

u and v being determinate functions of x, y, z, p, and q; and f being an arbitrary functional symbol. From these first integrals, singly or in combination, the second integral involving two arbitrary functions is obtained by a subsequent integration.

Now this procedure involves the assumption that the proposed equation admits of a first integral of the form (4). But such is not always the case. There exist primitive equations involving two arbitrary functions, from which by proceeding to a second differentiation both functions may be eliminated and an equation of the form (2) obtained, but from which it is impossible to eliminate one function only so as to lead to an intermediate equation of the form (4). Especially this happens if the primitive involve an arbitrary function and its derived function together. Thus the primitive

2= $ (y + x) + ¥ (y − x) − x {☀' (y + x) — y' (y − x)}...(5),

leads to the partial differential equation of the second order

but not through an intermediate equation of the form (4).

It is necessary therefore, not only to consider the case in which the assumed condition is satisfied, but also to notice

what has been done in those cases which do not at present fall under the dominion of any known method.

Genesis of the Equation.

2. PROP. I. A partial differential equation of the first order of the form u =ƒ (v), or its symmetrical equivalent, F(u, v) = 0,

in which u and v are any functions of x, y, z, p, q, always leads to a partial differential equation of the form

For, differentiating the proposed first integral with respect to x, and with respect to y, we have