DIFFERENTIAL tisfvinCT F = 0. In general, if functions G,H, independent of F, be obtained such that the equations F = 0, G = &, II = c represent an integral for all values of the constants b, c,4 these equations are said to constitute a complete integral. Then co elements satisfying F = 0 are known, and in fact every other form of integral can be obtained without further integrations. In the foregoing discussion of the differential equations ot a characteristic chain, the denominators • • may be supposed to be modified in form by means of F = 0 in any way conducive to a simple integration. In the immediately following explanation of ideas however, we consider indifferently all equations F = constant; when a function of xyzpq is said to be zero, it is meant that this is so identically, not in virtue of F = 0 ; in other words, we consider the integration of F = a, where a is an arbitrary constant. In the theory o°f linear partial equations we have seen that the integration of the of the characteristic chains, from which, Operations as ]UIS equations just seeri) that of the equation F = a follows necessary at oncej Would be involved in completely integrating ior single linear homogeneous partial differential Integra- the eqUati0n of the first order [F/] = 0, where the notation tion of _ .g t|iat eXpiaine(l above under Contact Transformations. P a ~ ' One obvious integral is/= F. Putting F — a, where a is arbitrary, and eliminating one of the independent variables, we can reduce this equation [F/] = 0 to one in four variables ; and so on. Calling, then, the determination of a single integral of a single homogeneous partial differential equation of the first order in n independent variables, an operation of order n — 1, the characteristic chains, and therefore the most general integral of F = a, can be obtained by successive operations of orders 3, 2, 1. If, however, an integral of F=a be represented by F-a, G=b, H=c, where b and c are arbitrary constants, the expression of the fact that a characteristic chain of F = a satisfies dG = 0, gives [FG] = 0 ; similarly, [FH] = 0 and [GH] = 0, these three relations being identically true. Conversely, suppose that an integral G, independent of F, has been obtained of the equation [F/] = 0, which is an operation of order three. Then it follows from the identity mm + + [^[/^>]]=fCM + d£m + ^zm before remarked, by putting <p=F, ^=G, and then [!/] = A(/), [G/] = B(/), that AB(/)-BA(/=^B(/)-^A(/), so that the two linear equations [F/] = 0, [G/] = 0 form a complete system; as two integrals F, G are known, they have a common integral H, independent of F, G, determinable by an operation of order one only. The three functions F, G, H thus identically satisfy the relations [FG] = [GH]=[FH] = 0. The oo2 elements satisfying F=a, G=b, H—Cj wherein a, b, c ai’e assigned constants, can then be seen to constitute an integral of F = a. For the conditions that a characteristic chain of G=b issuing from an element satisfying F=ct, G = 5, H = c should consist only of elements satisfying these three equations are simply [FG] = 0 [GH] = 0. Thus, starting from an arbitrary element of (F = a, G = 6, H = c), we can single out a connectivity of elements of (F = a, G = b, H = c) forming a characteristic chain of G — & ; then the aggregate of the characteristic chains of F = <x issuing from the elements of this characteristic chain of G = 5 will be a connectivity consisting only of elements of (F = a, G=b, H = c), and will therefore constitute an integral of F = a ; further, it will include all elements of (F = a, G — b, H = c). This result follows also from a theorem given under Contact Transformations, which shows, moreover, that though the characteristic chains of F = a are not determined by the three equations F = a, G = b, H = c, no further integration is now necessary to find them. By this theorem, since identically [FG] = [GH] = [FH] = 0, we can find, by the solution of linear algebraic equations only, a non-vanishing function <r and two functions A, C, such that dG - KdF - CrfH = a(dz -pdx - qdy) ■, thus all the elements satisfying F = a, G = &, Fi = c, satisfy dz=pdx + qdy and constitute a connectivity, which is therefore an integral of F = a. While, further, from the associated theorems, F, G, H, A, C are independent functions and i[FC] — 0. Thus C may be taken to be the remaining integral independent of G, H, of the equation [F/'] = 0, whereby the characteristic chains are entirely determined. When we consider the particular equation F = 0, neglecting the case when neither p nor q enters, and supposing p to enter, we may express p from F = 0 in terms of xyzq, and then eliminate it from all other equations. Then instead of the equation [F/] = 0, we have, if F = 0 give p = p(xyzq), the equation fi/= ^ ) _ oj m oreover obtainable by omit+ + „^ dqdy 1 dz) dy ^ dz ) dq ting the term in in [# - Q, /] = 0. Let x0yyz0q0 be values about dp which the coefficients in this equation are developable, and let y, w be the principal solutions reducing respectively to z, y and q when x—x0. Then the equations p=^, $=z0, y=y<» o} — q0
EQUATIONS
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represent a characteristic chain issuing from the element x0y0z0f 0q0 ; we have seen that the aggregate of such chains issuing from the elements of an arbitrary chain satisfying dz0 -p0dx0 single q0dy0=zQ constitute an integral of the equation p = 'f. equatioa Let this arbitrary chain be taken so that x0 is constant; aad then the condition for initial values is only dzg - q0dy0— pfaffjaa 0, and the elements of the integral constituted by formuia. the characteristic chains issuing therefrom satisfy d£- ^OHwdT) = 0. Hence this equation involves dz - pdx - qdy = 0, or we have dz — ipdx — qdy = (r{d£ — indy), where cr is not zero. Conversely, the integration oip—xf is, essentially, the problem of writing the expression dz — pdx — qdy in the form <i(d'y — tody), as must be possible (from what was said under Pfafian Expressions). To integrate a system of simultaneous equations ot the first order X1 = a1 • • • Xr = ar in n independent variables aq • ■ xn and one dependent variable 3, we write p1 for dzdxx, &c., s tem of and attempt to find n + l-r further functions Z, Xr+1 equafjons • • Xn, such that the equations Z=a, Xi-=afi=l ■ • n) Qf the first involve dz -p1dxl - • • -pndxn=Q. By an argument or(jeri already given, the common integral, if existent, must be satisfied by the equations of the characteristic chains of any one equation Xj = ae-; thus each of the expressions [XiXj] must vanish in virtue of the equations expressing the integral, and we may without loss of generality assume that each of the corresponding r(r — ) expressions formed from the r given differential equations vanishes in virtue of these equations. The determination of the remaining n + l-r functions may, as before, be made to depend on characteristic chains, which in this case, however, are manifolds of r dimensions obtained by integrating tlxe equations [X,/] = 0 • • • [Xr/] = 0 ; or having obtained one integral of this system other than Xj • • Xr, say Xr+1, we may consider the system [X1/] = 0 • • [Xr+1/] = 0, forwhich, again, we haveachoice ; and at any stage we may use Mayer’s method and reduce the simultaneous linear equations to one equation involving parameters ; while, if at any stage of the process we find some but not all of the integrals of the simultaneous system, they can be used to simplify the remaining work ; this can only be clearly explained in connexion with the theory of so - called function groups for which we have no space. One result arising is that the simultaneous system Pi = 0i, • • , pr=(pr, wherein Pi • • Pr are not involved in <px . . <pr, if it satisfies the ir(r-l) relations [>-<&, Pj-<pj] = 0, has a solution z — ^/(xl • • x„), p^d^jdx^ • •, 2)n — d-fjdxH, reducing to an arbitrary function, of xr+1 • • • xn only,, when x^ = xf • • xr =zxr° under certain conditions as to developability ; a generalization of the 7theorem for linear equations. The problem of integration of this S3 stem is, as before, to put dz — rpyixi — • • — <prdxr — j)rjr+lxr+^ - • • —pn dxn into the form a{d^-ur+fl£r+1 - • •- oind^n) ; and here f £r+1 • • £« wr+1 • • • wn may be taken, as before, to be principal integrals of a certain complete system of linear equations ; those, namely, determining the characteristic, chains. If L be a function of t and of the 2n quantities x1 • • XnXj • • xn, where ii denotes dxijdt, &c., and if in the equations= and so express ■ • xn in terms of t, xl- — we put Pi: .dL dxf dxi cZ2Ii XnPi • • Pn, assuming that the determinant of the quantities ^ ^ is not zero ; if, further, H denote the function of t, xv- ■ •• pn, numerically equal to i+x^ + • • +pnxn - L, it is easy p tl ns to prove tYiaXdpildt— — dFUdxi, dxijdt = dGIdpi. These ^ so-called canonical equations form part of those for the dynam!cs. characteristic chains of the single partial equation dzjdt + H(te1 • • xn, dz/dxi, • • • , dzldxf = 0, to which then the solution of the original equations for xx • • xn can be reduced. It may be shown (1) that if z=p(tx1 • • x„c1 • • c„) + c be a complete integral of this equation, then pi^dx/z/dx,, drfjdci — Ci are 2n equations giving the solution of the canonical equations referred to, where Cj • • c„ and e1 ■ ■ en are arbitrary constants ; (2) that if Xi=Xi(tx • • Pn°), Pi — Fftxf • • p°n) be the principal solutions of the canonical equations for t=t°, and w denote the result of substituting these values in y)YdPljdpx + • • • +pndlljdpn-Fl, and t O = / oidt, where, after integration, O is to be expressed as a function of t, x1 • • xnxf • • xn°, then z = fl + z° is a complete integral of the partial equation. A system of differential equations is said to allow a certain continuous group of transformations (see Groups) when the introduction for the variables in the differential equations of the new variables arising in the general finite equations of the group leads, for all values of the parameters of the group, to the same differential equations in the new variables. It would be interesting
