Page:EB1911 - Volume 08.djvu/249

identities [ZX_i] = 0, [X_iX_j] = 0. And the further identities

$[\mathrm {Z} \sigma ]{=}\sigma {\frac {d\mathrm {Z} }{dz}}-\sigma ^{2},[\mathrm {X} _{i}\sigma ]{=}\sigma {\frac {d\mathrm {X} _{i}}{dz}},[\mathrm {P} _{i}\sigma ]{=}{\frac {d\mathrm {P} _{i}}{dz}}$

are also verified. Conversely, if Z, x₁, ... X_n be independent functions satisfying the identities [ZX_i] = 0, [X_iX_j] = 0, then σ, other than zero, and P₁, ... P_n can be uniquely determined, by solution of algebraic equations, such that

dZ − P₁dX₁ − ... − P_ndX_n＝σ(dz − p₁dx₁ − ... − p_ndx_n).

Finally, there is a particular case of great importance arising when σ = 1, which gives the results: (1) If U, X₁, ... X_n, P₁, ... P_n be 2n + 1 functions of the 2n independent variables x₁, ... x_n, p₁, ... p_n, satisfying the identity

dU + P₁dx₁ + ... + P_ndX_n＝p₁dx₁ + ... + p_ndx_n,

then the 2n functions P₁, ... P_n, X₁, ... X_n are independent, and we have

(X_iX_j)＝0, (X_iU)＝δX_i, (P_iX_i)＝1, (P_iX_j)＝0, (P_iP_j)＝0, (P_iU) + P_i＝δP_i,

where δ denotes the operator p₁d/dp₁ + ... + p_nd/dp_n; (2) If X₁, ... X_n be independent functions of x₁, ... x_n, p₁, ... p_n, such that (X_iX_j) = 0, then U can be found by a quadrature, such that

(X_iU)＝δX_i;

and when X_i, ... X_n, U satisfy these 1/2n(n + 1) conditions, then P₁, ... P_n can be found, by solution of linear algebraic equations, to render true the identity dU + P₁dX₁ + ... + P_ndX_n = p₁dx₁ + ... + p_ndx_n; (3) Functions X₁, ... X_n, P₁, ... P_n can be found to satisfy this differential identity when U is an arbitrary given function of x₁, ... x_n, p₁, ... p_n; but this requires integrations. In order to see what integrations, it is only necessary to verify the statement that if U be an arbitrary given function of x₁, ... x_n, p₁, ... p_n, and, for r < n, X₁, ... X_r be independent functions of these variables, such that (XσU) = δXσ, (XρXσ) = 0, for ρ, σ = 1 ... r, then the r + 1 homogeneous linear partial differential equations of the first order (Uƒ) + δƒ = 0, (Xρƒ) = 0, form a complete system. It will be seen that the assumptions above made for the reduction of Pfaffian expressions follow from the results here enunciated for contact transformations.

We pass on now to consider the solution of any partial differential equation of the first order; we attempt to explain certain ideas relatively to a single equation with any number of independent variables (in particular, an ordinary equation of the first order with one independent Partial differential equation of the first order.variable) by speaking of a single equation with two independent variables x, y, and one dependent variable z. It will be seen that we are naturally led to consider systems of such simultaneous equations, which we consider below. The central discovery of the transformation theory of the solution of an equation F(x, y, z, dz/dx, dz/dy) = 0 is that its solution can always be reduced to the solution of partial equations which are linear. For this, however, we must regard dz/dx, dz/dy, during the process of integration, not as the differential coefficients of a function z in regard to x and y, but as variables independent of x, y, z, the too great indefiniteness that might thus appear to be introduced being provided for in another way. We notice that if z = ψ(x, y) be a solution of the differential equation, then dz = dxdψ/dx + dydψ/dy; thus if we denote the equation by F(x, y, z, p, q,) = 0, and prescribe the condition dz = pdx + qdy for every solution, any solution such as z = ψ(x, y) will necessarily be associated with the equations p = dz/dx, q = dz/dy, and z will satisfy the equation in its original form. We have previously seen (under Pfaffian Expressions) that if five variables x, y, z, p, q, otherwise independent, be subject to dz − pdx − qdy = 0, they must in fact be subject to at least three mutual relations. If we associate with a point (x, y, z) the plane

Z − z＝p(X − x) + q(Y − y)

passing through it, where X, Y, Z are current co-ordinates, and call this association a surface-element; and if two consecutive elements of which the point(x + dx, y + dy, z + dz) of one lies on the plane of the other, for which, that is, the condition dz = pdx + qdy is satisfied, be said to be connected, and an infinity of connected elements following one another continuously be called a connectivity, then our statement is that a connectivity consists of not more than ∞² elements, the whole number of elements (x, y, z, p, q) that are possible being called ∞⁵. The solution of an equation F(x, y, z, dz/dx, dz/dy) = 0 is then to be understood to mean finding in all possible ways, from the ∞⁴ elements (x, y, z, p, q) which satisfy F(x, y, z, p, q) = 0 a set of ∞² elements forming a connectivity; or, more analytically, finding in all possible ways two relations G = 0, H = 0 connecting x, y, z, p, q and independent of F = 0, so that the three relations together may involve

dz＝pdx + qdy.

Such a set of three relations may, for example, be of the form z = ψ(x, y), p = dψ/dx, q = dψ/dy; but it may also, as another case, involve two relations z = ψ(y), x = ψ₁(y) connecting x, y, z, the third relation being

ψ′(y)＝pψ′₁(y) + q,

the connectivity consisting in that case, geometrically, of a curve in space taken with ∞¹ of its tangent planes; or, finally, a connectivity is constituted by a fixed point and all the planes passing through that point. This generalized view of the meaning of a solution of F = 0 is of advantage, moreover, in view of anomalies otherwise arising from special forms of the equation Meaning of a solution of the equation. itself. For instance, we may include the case, sometimes arising when the equation to be solved is obtained by transformation from another equation, in which F does not contain either p or q. Then the equation has ∞² solutions, each consisting of an arbitrary point of the surface F = 0 and all the ∞² planes passing through this point; it also has ∞² solutions, each consisting of a curve drawn on the surface F = 0 and all the tangent planes of this curve, the whole consisting of ∞² elements; finally, it has also an isolated (or singular) solution consisting of the points of the surface, each associated with the tangent plane of the surface thereat, also ∞² elements in all. Or again, a linear equation F = Pp + Qq − R = 0, wherein P, Q, R are functions of x, y, z only, has ∞² solutions, each consisting of one of the curves defined by

dx/P＝dy/Q＝dz/R

taken with all the tangent planes of this curve; and the same equation has ∞² solutions, each consisting of the points of a surface containing ∞¹ of these curves and the tangent planes of this surface. And for the case of n variables there is similarly the possibility of n + 1 kinds of solution of an equation F(x₁, ... x_n, z, p₁, ... p_n) = 0; these can, however, by a simple contact transformation be reduced to one kind, in which there is only one relation z′ = ψ(x′₁, ... x′_n) connecting the new variables x′₁, ... x′_n, z′ (see under Pfaffian Expressions); just as in the case of the solution

z＝ψ(y), x＝ψ₁(y), ψ′(y)＝pψ′₁(y) + q

of the equation Pp + Qq = R the transformation z′ = z − px, x′ = p, p′ = −x, y′ = y, q′ = q gives the solution

z′＝ψ(y′) + x′ψ₁(y′), p′＝dz′/dx′, q′＝dz′/dy′

of the transformed equation. These explanations take no account of the possibility of p and q being infinite; this can be dealt with by writing p = −u/w, q = −v/w, and considering homogeneous equations in u, v, w, with udx + vdy + wdz = 0 as the differential relation necessary for a connectivity; in practice we use the ideas associated with such a procedure more often without the appropriate notation.

In utilizing these general notions we shall first consider the theory of characteristic chains, initiated by Cauchy, which shows well the nature of the relations implied by the given differential equation; the alternative ways of carrying out the necessary integrations are suggested by considering Order of the ideas.the method of Jacobi and Mayer, while a good summary is obtained by the formulation in terms of a Pfaffian expression.

Consider a solution of F = 0 expressed by the three independent equations F = 0, G = 0, H = 0. If it be a solution in which there is more than one relation connecting x, y, z, let new variables x′, y′, z′, p′, q′ be introduced, as before explained under Pfaffian Expressions, in which z′ is of the formCharacteristic chains.

z′＝z−p₁x₁−... −p_sx_s (s＝1 or 2),

so that the solution becomes of a form z′ = ψ(x′y′), p′ = dψ/dx′, q′ = dψ/dy′, which then will identically satisfy the transformed equations F′ = 0, G′ = 0, H′ = 0. The equation F′ = 0, if x′, y′, z′ be regarded as fixed, states that the plane Z − z′ = p′(X − x′) + q′(Y − y′) is tangent to a certain cone whose vertex is (x′, y′, z′), the consecutive point (x′ + dx′, y′ + dy′, z′ + dz′) of the generator of contact being such that

$dx'\left/{\frac {d\mathrm {F} '}{dp'}}\right.=dy'\left/{\frac {d\mathrm {F} '}{dq'}}\right.=dz'\left/\left(p'{\frac {d\mathrm {F} '}{dp'}}+q'{\frac {d\mathrm {F} '}{dq'}}\right)\right..$

Passing in this direction on the surface z′ = ψ(x′, y′) the tangent