2ndly. Jacobi shews how by the aid of such derived integrals of the first equation of the system a common integral of the first and second equation may be found, and how from this integral and its derived series a common integral of the first three equations of the system may be found, and so on, until a common integral of the entire system has been as it were built up out of previous integrals of less general application.

......

Let $, $', $", Φ, (-1) represent a series of independent integrals of the equation A,P=0, of which 4 is the root integral, and the rest are derived from it by successive applications of the operation denoted by A,, so that

also let A," be not a new integral but a function of

Now $, $',...... $-1) being particular integrals of A ̧P=0, the function F'(, p', ...... p-1)) will also be an integral of the same equation irrespectively of its form. Let us inquire whether the form of the function can be so determined as to render it also an integral of the second equation A,P=0.

By the principles of the Differential Calculus this equation assumes the form

But A‚$ = &', A‚p' = 4", ...... A ̧$(μ-2) = $(−1) ;

lastly, A-1) may by hypothesis be expressed in the form ƒ (P, $',...... $(-1)). Thus the equation to be satisfied is

th

Now the integration of this system may be made to depend upon that of an ordinary differential equation of the (1)" degree between the two variables (-1) and p.

Differentiating the last equation with respect to p, and attend

ing to the former ones, we shall be able to express

d2 flu-1)

dp*

in

terms of the variables p, p, ...-1). Proceeding with this in the same way and continuing the process we shall be able to express the series of differential coefficients

.....

bad notation

in terms of p, q',
-1. From these μ-1 equations,
eliminating p', p",..... p-2), we shall have a final equation.
...-2),

between

that is, a differential equation of the (u-1)th order between and

-1).

The complete integral of this equation will be of the form

pm-1) = f (p, c1, Ca, Cμ-1).

.....

Differentiating this μ-2 times in succession with respect to o, and continually substituting for the differential coefficients of p-1) their values as before assigned in terms of

we shall have a system of μ-1 equations connecting the above variables with the constants C, C,..... C-1. Finally, solving these equations with respect to the constants, we shall possess the integrals required in the form

and each of these will be a common integral of the first two equations of the given system (1).

[On the back of a page of the manuscript the following paragraph occurs, which seems to have been intended as a simplification of the preceding argument which begins with "The complete integral.”]

Suppose that a first integral of the equation can be found. Its form will be

Substitute in this for the differential coefficients of (-1) their values before assigned in terms of p, p, p,...), and we have an integral of the system (3), and therefore a common integral of the first two equations of the system (1).

[We now return to the place at which we inserted a paragraph.]

Just in the same way Jacobi deduces a common integral of the first three equations of the system (1). For representing

any one of the first members of the above system by, and deriving thence the new independent integrals Ay, Ay,... he substitutes an arbitrary function of these for P in the equation

A,P=0.

It is evident that the solution of the partial differential equation so found will again be reducible to that of an ordinary differential equation between two variables. And so the process is carried on till all the equations are satisfied.

2. The above remarkable process was developed by Jacobi in connexion with the theory of non-linear partial differential equations of the first order. In that particular connexion it admits of certain reductions tending to diminish the order of the differential equations to be integrated. But these do not affect the general principle of the method. It was in this special form that the theory of the solution of simultaneous linear partial differential equations originated. Jacobi does not consider the theory of equations in which the condition (2) is not satisfied; but the language in which he refers to the condition shews that he had speculated upon the general problem-and it is difficult to conceive that he should have meditated upon it and not arrived at its complete solution.

[The manuscript here gives the first two words of the passage from Jacobi's memoir which is quoted in the Philosophical Transactions for 1863, page 486.]

CHAPTER XXVII.

OF NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER.

1. IN treating the present subject we shall first consider that class of non-linear partial differential equations of the first order which involves two independent variables, and then proceed to the general theory. The reason for this procedure is that the particular theory, though of course included in the general one, rests upon a somewhat simpler basis, and it was in fact developed by the labours of Lagrange and Charpit long before the general theory was known. The latter we owe to the independent researches of Cauchy and Jacobi.

[Here the manuscript refers to the matter contained in Chap. XIV. Arts. 7 to 12 inclusive; and then passes on to the general theory.]

...

the number of arbitrary constants a, a, ... an involved being equal to the number of the independent variables x1, x2, Xn, we obtain by differentiation and elimination of the constants a partial differential equation of the first order. Of this the proposed equation is said to constitute a complete primitive.