which is the determinant form of the result affirmed in the proposition. The second of the above forms of demonstration seems to be preferable to the first, in that it rests only upon the consideration of the one general form of the function F. I have, however, given the two proofs, chiefly in order to illustrate an important remark, viz. that, in nearly all general researches connected with partial differential equations of the first order, two modes of procedure, the one involving the use of differentials, the other that of differential coefficients, may be employed, and that between the forms to which these respective modes give rise, a certain law of reciprocity will be found to exist. The theory of the solution of the partial differential equation follows immediately from that of its genesis. If we represent by the integrals of the system of ordinary differential equations (2) a solution of the given partial differential equation will be represented by (1). That this will be also the most general solution may be shewn by the argument of Art. 1. For if w = 0 represent any solution, then since dw + Pn dz dxn 0, dw dw +X + R -2 dx2 ... dz 0, ... + Xn ... dz dun dz = = 0, ... eliminating X, X2, ... X, R we obtain a result which expresses that the functional determinant of w, u,,... un with respect to the original variables is virtually 0. Whence w is a function of u1, u,,... u,, and the proposed solution is included in the one to which the above method of solution leads. That method may therefore be stated in the following Rule. RULE. To integrate the linear partial differential equation F(u1, u2,... u2) = 0 will be the general integral sought. [The general observations were intended to follow Chap. XIV. Art. 6.] General observations. 4. The relation which exists between a proposed linear partial differential equation and its auxiliary system of ordinary differential equations should be carefully studied. While it is proper to say as above that the general integral of the one requires the knowledge of all the integrals of the other, it is also proper to describe that general integral simply as the most general form under which an integral of the auxiliary system can appear. If are integrals of that system, then F(u1, u2,... un) = A is the one general form of an integral of that system, and due regard being had to the arbitrariness of F, this is equi valent to 5. The form which the auxiliary system assumes when the given partial differential equation is deficient in any of its terms should be noticed. If X1 = 0, the auxiliary equation And thus, if X1, X,,... X, vanish, the given equation being dz the auxiliary system will be and the integrals of this system being of the form the general solution of the given equation will be F(x,,.. Xr, Ur+1) un) = 0. ...... This conclusion would follow also from the principle laid down in Chap. xiv. Art. 2. Linear partial differential equations in which the absolute term is wanting, and which are therefore of the form may be termed homogeneous. As in this case one of the auxiliary equations is dz = 0, the general integral will be F(u1, u2, Un-1, 2) = 0, ...... u1, u,,......u being found by the integration of the remaining auxiliary equations When X1, X2, .... X do not contain z, the solution is best exhibited in the form 6. Every linear partial differential equation can be converted into a homogeneous one containing one additional variable. For it is shewn in Art. 3, that if u = 0 be any integral of a homogeneous equation with a new variable. From the general integral of this equation, that of the former one may be deduced by making u = 0. 7. The solution of partial differential equations is sometimes facilitated by introducing a new system of independent variables. The actual transformation is greatly facilitated by the following symbolical theorem. then, if y1, 3, ...... y, be a new system of independent variables given in expression as functions of the old ones, the transformed equation will be For, regarding z as a function of y,, Y., ...... Yn, we have = dx dz dy, dz dy, + + dz dyn; ... dy dx dy, dx dyn dxn whence, substituting in the given equation we find, as the dz total coefficient of dy,' the expression or symbolically, Ay,; and so on for the other coefficients. The result then is It remains only after calculation of Ay, Ay,, ......▲yn, as functions of x,, x,,......, to express these functions and X in terms of y1, Y2) ...... Yn n [It appears from the manuscript that an example was to have been supplied here.] |