Page:EB1911 - Volume 08.djvu/250

 plane of the surface at this consecutive point is (p′ + dp′, q′ + dq′), where, since F′(x′, y′,, d/dx′, d/dy′)＝0 is identical, we have dx′ (dF′/dx′ + p′dF′/dz′) + dp′dF′/dp′＝0. Thus the equations, which we shall call the characteristic equations, $dx'\left/\frac{d\mathrm{F}'}{dp'}\right.=dy'\left/\frac{d\mathrm{F}'}{dq'}\right.=dz'\left/\left(p'\frac{d\mathrm{F}'}{dp'} + q'\frac{d\mathrm{F}'}{dq'}\right)\right.=dp'\left/\left( - \frac{d\mathrm{F}'}{dx'} - p'\frac{d\mathrm{F}'}{dz'}\right)\right. =dq'\left/\left( - \frac{d\mathrm{F}'}{dy'} - q'\frac{d\mathrm{F}'}{dz'}\right)\right.$ are satisfied along a connectivity of ∞¹ elements consisting of a curve on z′＝(x′, y′) and the tangent planes of the surface along this curve. The equation F′＝0, when p′, q′ are fixed, represents a curve in the plane Z − z′＝p′(X − x′) + q′(Y − y′) passing through (x′, y′, z′); if (x′ + x′, y′ + y′, z′ + z′) be a consecutive point of this curve, we find at once $\delta x'\left(\frac{d\mathrm{F}'}{dx'} + p'\frac{d\mathrm{F}'}{dz'}\right) + \delta y'\left(\frac{d\mathrm{F}'}{dy'}q'\frac{d\mathrm{F}'}{dz'}\right)=0;$ thus the equations above give x′dp′ + y′dq′＝0, or the tangent line of the plane curve, is, on the surface z′＝(x′, y′), in a direction conjugate to that of the generator of the cone. Putting each of the fractions in the characteristic equations equal to dt, the equations enable us, starting from an arbitrary element x′0, y′0, z′0, p′0, q′0, about which all the quantities F′, dF′/dp′, &c., occurring in the denominators, are developable, to define, from the differential equation F′＝0 alone, a connectivity of ∞¹ elements, which we call a characteristic chain; and it is remarkable that when we transform again to the original variables (x, y, z, p, q), the form of the differential equations for the chain is unaltered, so that they can be written down at once from the equation F＝0. Thus we have proved that the characteristic chain starting from any ordinary element of any integral of this equation F＝0 consists only of elements belonging to this integral. For instance, if the equation do not contain p, q, the characteristic chain, starting from an arbitrary plane through an arbitrary point of the surface F＝0, consists of a pencil of planes whose axis is a tangent line of the surface F＝0. Or if F＝0 be of the form Pp + Qq＝R, the chain consists of a curve satisfying dx/P＝dy/Q＝dz/R and a single infinity of tangent planes of this curve, determined by the tangent plane chosen at the initial point. In all cases there are ∞³ characteristic chains, whose aggregate may therefore be expected to exhaust the ∞4 elements satisfying F＝0.

Consider, in fact, a single infinity of connected elements each satisfying F＝0, say a chain connectivity T, consisting of elements specified by x0, y0, z0, p0, q0, which we suppose expressed as functions of a parameter u, so that U0＝dz0/du − p0dx0/du − q0dy0/du is everywhere zero on this chain; further, suppose that each of F, dF/dp, ..., dF/dx + pdF/dz is developable about each element of this chain T, and that T is not a characteristic chain. Then consider the aggregate of the characteristic chains issuing from all the elements of T. The ∞² elements, consisting of the aggregate of these characteristic chains, satisfy F＝0, provided the chain connectivity T consists of elements satisfying F＝0; for each characteristic chain satisfies dF＝0. It can be shown that these chains are connected; in other words, that if x, y, z, p, q, be any element of one of these characteristic chains, not only is dz/dt − pdx/dt − qdy/dt＝0, as we know, but also U＝dz/du − pdx/du − qdy/du is also zero. For we have $\frac{d\mathrm{U}}{dt} =\frac{d}{dt}\left(\frac{dz}{du} - p\frac{dx}{du} - q\frac{dy}{du}\right) - \frac{d}{du}\left(\frac{dz}{dt} - p\frac{dx}{dt} - q\frac{dy}{dt}\right)$|undefined $ =\frac{dp}{du}\frac{dx}{dt} - \frac{dp}{dt}\frac{dx}{du} + \frac{dq}{du}\frac{dy}{dt} - \frac{dq}{dt}\frac{dy}{du}$ which is equal to $\frac{dp}{du}\frac{d\mathrm{F}}{dp} + \frac{dx}{du}\left(\frac{d\mathrm{F}}{dx} + p\frac{d\mathrm{F}}{dz}\right) + \frac{dq}{du}\frac{d\mathrm{F}}{dq} + \frac{dy}{du}\left(\frac{d\mathrm{F}}{dy} + q\frac{d\mathrm{F}}{dz}\right) =- \frac{d\mathrm{F}}{dz}\mathrm{U}.$|undefined As dF/dz is a developable function of t, this, giving $\mathrm{U} =\mathrm{U}_{0}exp\left( - \int^{t}_{t_{0}}\frac{d\mathrm{F}}{dz}dt \right),$|undefined shows that U is everywhere zero. Thus integrals of F＝0 are obtainable by considering the aggregate of characteristic chains issuing from arbitrary chain connectivities T satisfying F＝0; and such connectivities T are, it is seen at once, determinable without integration. Conversely, as such a chain connectivity T can be taken out from the elements of any given integral all possible integrals are obtainable in this way. For instance, an arbitrary curve in space, given by x0＝(u), y0＝(u), z0＝(u), determines by the two equations F(x0, y0, z0, p0, q0)＝0, ′(u)＝p0′(u) + q0′(u), such a chain connectivity T, through which there passes a perfectly definite integral of the equation F＝0. By taking ∞² initial chain connectivities T, as for instance by taking the curves x0＝, y0 ＝, z0 ＝ to be the ∞² curves upon an arbitrary surface, we thus obtain ∞² integrals, and so ∞4 elements satisfying F＝0. In general, if functions G, H, independent of F, be obtained, such that the equations F＝0, G＝b, H＝c represent an integral for all values of the constants b, c, these equations are said to constitute a complete integral. Then ∞4 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 of a characteristic chain, the denominators dF/dp, ... 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 x, y, z, p, q 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 of linear partial equations we have seen that the integration of the equations of the characteristic chains, from which, as has just been seen, that of the equation F＝a follows at once, would be involved in completely integrating the single linear homogeneous partial differential equation of the first order [Fƒ]＝0 where the notation is that explained above under Contact Transformations. One obvious integral is ƒ＝F. Putting F＝a, where a is arbitrary, and eliminating one of the independent variables, we can reduce this equation [Fƒ]＝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 [Fƒ]＝0, which is an operation of order three. Then it follows from the identity [ƒ[]] + ƒ + ƒ＝dƒ/dz [] + d/dz [ƒ] + d/dz [ƒ] before remarked, by putting ＝F, ＝G, and then [Fƒ]＝A(ƒ), [Gƒ]＝B(ƒ), that AB(ƒ) − BA(ƒ)＝dF/dz B(ƒ) − dG/dz A(ƒ), so that the two linear equations [Fƒ]＝0, [Gƒ]＝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 ∞² elements satisfying F＝a, G＝b, H＝c, wherein a, b, c are 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＝a, G＝b, 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＝b, H＝c), we can single out a connectivity of elements of (F＝a, G＝b, H＝c) forming a characteristic chain of G＝b; then the aggregate of the characteristic chains of F＝a issuing from the elements of this characteristic chain of G＝b 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 and two functions A, C, such that dG − AdF − CdH＝(dz − pdz − qdy); thus all the elements satisfying F＝a, G＝b, H＝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 [FC]＝0. Thus C may be taken to be the remaining integral independent of G, H, of the equation [Fƒ]＝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 x, y, z, q, and then eliminate it from all other equations. Then instead of the equation [Fƒ]＝0, we have, if F＝0 give p＝(x, y, z, q), the equation $\Omega f =- \left(\frac{df}{dx} + \psi \frac{df}{dz}\right) + \frac{d\psi}{dq}\left(\frac{df}{dy} + q\frac{df}{dz}\right) - \left(\frac{d\psi}{dy} + q\frac{d\psi}{dz}\right)\frac{df}{dq}=0,$ moreover obtainable by omitting the term in dƒ/dp in [p −, ƒ]＝0. Let x0, y0, z0, q0, be values about which the coefficients in this equation are developable, and let  be the principal solutions reducing respectively to z, y and q when x＝x0. Then the equations p＝, ＝z0, ＝y0, ＝q0 represent a characteristic chain issuing from the element x0, y0, z0, 0, q0; we have seen that the aggregate of such chains issuing from the elements of an arbitrary chain satisfying

dz0＝p0dx0 − q0dy0＝0 constitute an integral of the equation p＝. Let this arbitrary