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any one of the first members of the above system by y, and deriving thence the new independent integrals 44, 4, 4,... he substitutes an arbitrary function of these for P in the equation
4 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.]
OF NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF THE
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.]
2. Given an equation of the form
an), the number of arbitrary constants ay, A2, An involved being equal to the number of the independent variables x, X, ... Lin, 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.
an) = aq
The form of the above process which it seems best, as throwing light upon the inverse problem of deducing the complete primitive from the partial differential equation, to employ, is the following. Let the given primitive, solved with respect to one of the arbitrary constants az, be presented in the form fla, xn, 2, а,,
(1). Differentiating with respect to each of the independent variables we have a system of n equations of the forms fi loc, Xn, Z, P, A,
an) = 0
(2). for (ime, Xn, Z, Png Ag, an) = 0 These n equations enable us first to eliminate the n-1 constants ag, ...... Am, and so deduce the partial differential
An equation sought in the form Fi (og, ... Xm, 2, P1, ... Pr) = 0 ....
........ (3); secondly to determine the n-1 constants as functions of Xn Xn, %, Pio
in the forms F(C Ung 2, Pay
(4). Xn, Z, P, ...
As the system formed of these n-1 equations, together with the previous one, is merely another form of the system (2) obtained by directly differentiating the primitive, it follows that if from these equations we deduce the values of P, ... Pre
• Pn as functions of X, ... Xn, Aqr.
An, and substitute them in the equation dz=p,dx, +p,dx, + ... + Podxm .........
prdxn (5), they will render that equation integrable, and its integral will be the complete primitive (1), the constant a, being regained by integration.
B. D. E. II.
Examining the system (3), (4) we see that the first members of all the equations which it contains are functions of ity,
Un, z, Pue ... Pn, while the second members are constants. The question then arises, What mutual connexion exists among these functions in virtue of which they yield values of P, ... Pm, which render the equation (5) integrable?
The answer to this question must involve the entire theory of the solution of partial differential equations of the first order, so far as relates to the determination of a complete primitive. Given a partial differential equation of the form (3) it is evident that if we can construct a system of associated equations (4) possessing the character above described, the final value of obtained by integration of (5) will both satisfy the given equation and contain the requisite number of arbitrary constants. It does not follow from this that it will be the only complete primitive, but it will be a complete primitive.
Pn) = 6
3. The relation sought is expressed in the following · Proposition : PROPOSITION. If
F(x, ... , 2, P1, ...
Pn) = a,
Xn, Z, Po represent any two out of a system of n independent equations such that the values of Pa, · Pn, thence determined would make the equation
dz=p,dx, +p,dx, +... + pndxn integrable, then the first members of these equations being represented for simplicity by Fand 0, the condition
(/dF dFI dΦ dF do dΦ
dz the summation extending to all values of i, from 1 to n inclusive, will be satisfied identically.
+ Pdf)} = 0,
Reciprocally, if the above condition be satisfied identically for each binary combination of functions in the proposed system of equations, and if these functions be independent, then the values of P... Pre as functions of x,, ... Xm, z, which they yield,
,, will make the equation
dz =p, dx, +p,dx, +... +Pndxn integrable.
It will be convenient to begin with the particular case in which the proposed equations do not explicitly contain 2, the
Differentiating with respect to Xi, and regarding P1, ... Pn as functions of the independent variables, we have
the summation with respect to j extending from j=1 to j=n inclusive.
do From the first of equations (7) multiplied by