from the above system combined with dP dP dy + dz + da dy dz dP dP dt = 0, dP dz and equate to 0 the coefficient of in the result. We find dz - (t+3x2) dx - ydy — xdt = 0, An arbitrary function of the first member of this equation is the general value of P. [It appears from the manuscript that another example was to have been added here.] CHAPTER XXVI. HOMOGENEOUS SYSTEMS OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS. 1. THE theory of homogeneous systems of linear partial differential equations in which when expressed in the symbolic form is for all combinations represented by i and j satisfied in virtue of the constitution of the symbols A, A,;, forms the subject of important researches by Jacobi (Nova Methodus... Crelle's Journal, Vol. LX. p. 1). The following are the most important of his results. 1st. An integral of any one equation of the system being found, other integrals of the same system may be obtained without integration, by a process of derivation founded upon the condition (2). Let be an integral of the first equation of the system. Then is the equation identically satisfied. Δφ= 0 Also the condition (2) being satisfied in virtue of the constitution of the symbols, we have (A;A; — A‚A;) &=0; and in particular, making i = 1, and separating the terms, It appears from this that A,, if it do not reduce to a constant, is an integral of the first equation 40=0, and, if it prove to be not a mere function of p, a new integral. This process may be repeated upon the new integral with a similar alternation of results. It will be evident from this that if we confine our attention to the two equations and will AP=0, AP=0, suppose, as before, & to be an integral of the first, then Δ.Φ, Δ. (ΔΦ), Δ. Δ. (ΔΦ)},... or, as these may be expressed, Δ.Φ. Δ. Φ, Δ. Φ. ... be also integrals of the first equation; and this process of derivation may be continued until we arrive at an integral A," which is not independent, but is expressible as a function of prior integrals Δ.Φ, Δ. Φ.......Δ. -φ, and, sooner or later, such a result must present itself, since the number of independent integrals is finite. It is further seen that the most general symbolic form of an integral derivable from the root integral o is α, B, A‚aA ̧3......A", ......., being positive integers. The above remarkable theorem was in some degree anticipated by the researches of Poisson. 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 $, $', o", Φ, (-1) represent a series of independent integrals of the equation A,P=0, of which is the root integral, and the rest are derived from it by successive applications of the operation denoted by ▲2, so that 27 also let ▲," be not a new integral but a function of Now p, p,...... 4-1) being particular integrals of ▲ ̧P=0, the function F(4, $', ...... -i) 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 lastly, A-1) may by hypothesis be expressed in the form ƒ (Þ, p',...... $-1)). Thus the equation to be satisfied is Now the integration of this system may be made to depend upon that of an ordinary differential equation of the (μ — 1) th degree between the two variables (-1) and 4. Differentiating the last equation with respect to p, and attend ing to the former ones, we shall be able to express d2 (4-1) 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 in terms of p, p', -1. From these μ-1 equations, eliminating ', ",..... p-2), we shall have a final equation that is, a differential equation of the (1)th order between and -1). |