Now the idea upon which Jacobi's later methods rest is that of directly solving the different systems of linear partial differential equations flowing from the general condition (1), not of solving, as in Cauchy's method, one of those equations and then limiting that solution by conditions which virtually involve the satisfaction of the others.

It is evident that the entire series of

n (n - 1)

2

conditions (1) will be satisfied if we determine F to satisfy the single equation

[FF]=0,

then F to satisfy the system of two simultaneous partial differential equations

[F1F,]=0, [F,F]=0,

then F to satisfy the system of three simultaneous partial differential equations

[F,F]=0, [F,F]=0, [FF]=0,

and so on, until finally F, is determined by the solution of the 'system of n-1 partial differential equations

[F,F„]=0, [F,F„] = 0,

[FF]=0.

Now all these are particular cases of the general problem of determining a function P which shall satisfy simultaneously the equations

[F,P]=0, [F2P] = 0, [FP]=0. (2)

......

F, F2, ... F being given functions between each pair of which the equation

[F;F;] = 0

is identically satisfied. Here P will represent in succession the series F, F, ... F.

The given system is one of homogeneous linear partial differential equations. It belongs to the class of systems the

general theory of which is discussed in Chap. xxvI. But it is not necessary to apply the theory in its general form. We need only a single integral; for a single value of each of the functions F, F, ... F, suffices in combination with the given value of F for the determination of a complete primitive. Now it may be shewn that the system is of the class discussed in Chapter XXVI. If expressed symbolically in the form

the condition

▲ ̧P=0, A‚P= 0, ..... A„P = 0,

(A;A; — A;A;) P=0,

will be identically satisfied. Hence Jacobi's method for the treatment of systems of this kind may be applied.

That the system is of the kind asserted is a consequence of the following proposition.

PROPOSITION. If the equations

[uP]=0, [vP] = 0

are expressed in the symbolic form

dv

dP

dpi

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

dx;

du

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

or

Lin (du d2v

du d'v dv d'u dv d'u

+ dx dpdx, dp, dxdxdx, dpdx,

dp, dxdx;

Hence (AA' — A'A) P = Σjn (d[uv] dP _ d[uv] dP

dx, dp

= [[uv] P],

whence the Proposition is established.

dp, dx;

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

But [FF]=0 by the conditions given; hence the condition (AA, — 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

Rr + Ss+ Tt = V.......

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

(2),

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).