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 4,P=0, 4,P=0, ...... AP=0 ... AMP= 0 ......... (1), the condition (4:4; – 4,4:) P=0.......... .(2) is for all combinations represented by i and j satisfied in virtue of the constitution of the symbols Ai, ;, 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 o be an integral of the first equation of the system. Then is the equation Δ.φ = 0 identically satisfied. Also the condition (2) being satisfied in virtue of the constitution of the symbols, we have (A,A; – 4;4;) $=0; and in particular, making i=1, and separating the terms, 4,4,4 – 4,4,$=0, which reduces by a prior equation to Δ.Δ,φ = 0. It appears from this that 4,0, if it do not reduce to a constant, is an integral of the first equation 4,0=0, and, if it prove to be not a mere function of $, 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 4,P= 0, 4,P=0, and suppose, as before, $ to be an integral of the first, then will Δ,φ, Δ, (Δ,φ), Δ,{Δ, (Δ.Φ)}, ... 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 4,1$ which is not independent, but is expressible as a function of prior integrals Δ,φ, 4,φ,......Δ,μ-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 $ is 4,44,8......Amus, as B. ...... , 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 0, $', $", .....(-1) represent a series of independent integrals of the equation 4,P=0, of which ¢ is the root integral, and the rest are derived from it by successive applications of the operation denoted by A,, so that φ' = Δ,φ, pw-2) = 4,4-1%; also let A,“ be not a new integral but a function of a Now $,$, ......$I) being particular integrals of AP=0, the function F($,$, ...... $w-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 4,P=0. We have then to satisfy the equation Δ,F (φ, φ', Фw = (). By the principles of the Differential Calculus this equation assumes the form But Δφ = φ', Δ,φ' = φ", ...... Δηφία - 2) = φμ -1); ' = 6", - 2) = ) lastly, 4,6–1 ) may by hypothesis be expressed in the form f(0, Ø....... $–2)). Thus the equation to be satisfied is a linear partial differential equation of which the auxiliary system is ...... $" dфu -1 do do' dФ — 2) dфи – 1) (3). $ фе -1) f (ф, ф", ...... фw -1) Now the integration of this system may be made to depend upon that of an ordinary differential equation of the (u – 1)th degree between the two variables (-1) and 0. For we have do'_$" dф — 2) фw -1) φ' Differentiating the last equation with respect to $, and attending to the former ones, we shall be able to express d"Фu — 1) in dø terms of the variables 0, $', ...... 6-1). Proceeding with this in the same way and continuing the process we shall be able to express the series of differential coefficients dфи-1) фu — 1) du-18 (-1) dф dф° in terms of $,$', fW-1). From these u-1 equations, eliminating $', $",..... 84-9), we shall have a final equation between dфи — 1) du-16-1) dфи-1 that is, a differential equation of the (u – 1)th order between $ and fl-1). dФ4 – 1) ..... The complete integral of this equation will be of the form $w-1)=f(, C2, Ca, ..... Cu-1). Differentiating this u-2 times in succession with respect to $, and continually substituting for the differential coefficients of fl-1) their values as before assigned in terms of $,$, ...... 61), we shall have a system of u – 1 equations connecting the above variables with the constants ci, Ca, ..... Cu-1. Finally, solving these equations with respect to the constants, we shall possess the integrals required in the form F ($,$', ..... fl-1)) = C1, F:-1($,$, 66-1)) = Cu-19 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 å 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 do(-1) F(ф, фи-1, d-->$(4) do Substitute in this for the differential coefficients of $(2) their values before assigned in terms of ¢, ', $",...piu1), and we have an integral of the system (3), and therefore a common integral of the first two equations of the system (1). , =C. dous [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 |