452
DIFFERENTIAL
a set of oo 2 elements forming a connectivity; or, more analytically, finding in all possible ways two relations G = 0, H = 0 connecting x, y, 2, 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 2 = ^(x, y), p = dij/Jdx, q = dj/ldy; but it may also, as another case, involve two relations z = ^(y), x = if/^y) connecting x, y, z, the third relation being t//(y) = pif/(y) + q, the connectivity consisting in that case, geometrically, of a curve in space taken with co 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 Meaning arising from special forms of the equation itself. ofasolu- For instance, we may include the case, sometion of the times arising when the equation to be solved equation. obtained by transformation from another equation, in which F does not contain either p or q. Then the equation has 002 solutions, each consisting of an arbitrary point of the surface F = 0 and all the 002 planes passing through this point; it also has co2 solutions, each consisting of a curve drawn on the surface F = 0 and all the tangent planes of this curve, the whole consisting of 002 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 go 2 elements in all. Or again, a linear equation F = Pjo + Qg - K, = 0, wherein P, Q, R are functions of x, y, z only, has co 2 solutions, each consisting of one of the curves defined by dx/F^ dy/Q, = dz/R taken with all the tangent planes of this curve; and the same equation has co 2 solutions, each consisting of the points of a surface containing co 1 of these curves and the tangent planes of this surface. And for the case of n variables there is similarly the possibility of % + 1 kinds of solution of an equation F(a:1 • • xrizp1 • • pn) = 0 ; these can, however, by a simple contact transformation be reduced to one kind, in which there is only one relation z = ^{x ■ • a/9l) connecting the new variables x • • x nz (see under Pfajjian Expressions) ; just as in the case of the solution z = i//(y), X^if/^y), ^'(y)=P^(y) + q of the equation Fp + Qc/ = Pi, the transformation z' = z-px, x! =p, p — -x, y =y, q =q gives the solution z' = ip(y') + x'xl/(yr)i V = dz jdx', q — dz jdy 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 ^heYdeas relati°ns implied by the given differential equation ; the alternative ways of carrying out the necessary integrations are suggested by considering 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 be introduced, as before explained under Pfaffian Charac- xyzp'q in which z is of the form z =z—p1x-l — teristic Expressions, • • -p (s = l or 2), so that the solution becomes of a chains. form z=p{x'y), p —df jdx, q =dffijdy', which then will identically satisfy the transformed equations F' = 0, G' = 0, H' = 0. The equation F/;:::::0, xyz be regarded as fixed, expresses that the plane Z-z'=/(X-aO + 2'(Y-2/') is tangent to a certain cone
EQUATIONS
whose vertex is x'y'z!, the consecutive point {x + dx, y + dz, z + dz) d dY' , ,1 of the generator of contact being such that dx dY' v= ylw=<bj V dp + dq / Passing in this direction on the surface 2' = ip(x'y') the tangent plane of the surface at this consecutive point is (p+dp, q+dq), where, since Y'(x, y', ip, dp jdx, dpjdy') = i) is identical, we have dx!{dY'jdx+p'dY'jdz) + dpdY'jdp =0. Thus, the equations, which we shall call the characteristic equations, dxd^= dy’l^=dz’l(p'^+q'~) = dp'l(-^ -p'^K I dp ^ I dq lY dpi * dq) 1 l dx r 1 dz ) = dq' I( are satisfied along a connectivity of oo1 eleI dy dz J ments consisting of a curve on z =p(x'y') and the tangent planes, of the surface along this curve. The equation F' = 0, whenp', q are fixed, represents a curve in the plane Z - z' =p'(X - x) + q(Y - y’)passing through xy'z ; if (x + dx, y' + dy', z + dz) be a consecutive point of this curve, we find at once dxl^Y-,+p'^YT + dy'(,^-TJr dx dz J dy ,dF_0 . tjius eqUations above give dxdp + dy'dq = 0, or thedz / tangent line of the plane curve, is, on the surface z =p{xy'), 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 0y'yz 0p yq 0, about which all the quantities F', dY'jdp, etc., occurring in the denominators, are developable, to define, from the differential equation F' = 0 alone, a connectivity of co 1 elements, which we call a characteristic chain; and it is remarkable that when we transform again to the original variables jxyzpq), 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 + Q»7 = R, the chain consists of a curve satisfying: dx/P = dy j Q = dzjR 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 00 3 characteristic chains, whose aggregate may therefore be expected to exhaust the co4 elements satisfying F = 0. Consider, in fact, a single infinity of connected elements eaclr satisfying F = 0, say a chain connectivity T, consisting of elements, specified by x0yyz0p0q0, which we suppose expressed as functions of a parameter u, so that U0 = dz0jdu -p0dx0jdu m Complete - qodyojdu is everywhere zero on this chain ; further, con-egra dFjdx + pdF jdz is suppose that each of F, d¥jdp, developable about each element of this chain T, and structed . that T is not a characteristic chain. Then consider ^jaracterthe aggregate of the characteristic chains issuing from jsijc all the elements of T. The 002 elements, consisting cliainS4 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 c£F = 0. It can be shown that these chains are connected;, in other words, that if xyzpq be any element of one of these characteristic chains, not only is dzjdt-pdxjdt- qdyjdt = Q, as we know, but also U =dzjdu -pdxjdu - qdyjdu is also zero. For , dll d (dz dx dy d (dz dx dy dp dx we Uve / dudi ^dt ^dt) dpdff^dq dp_(lq dy_^ wMch ig equal to dydF + dx (dF + «T dt du du dt dt du du dp du dx dz j dq dF dy (dF dF__dF..U. As L is a developable du dq +du dy+^dz) dz function of t, this, giving U = U0 exp{^~J
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 zinstance, an arbitrary curve in space, given by x0—d(u), ^o=0W> ^-rW> determines by the two equations F(x0yrjz0p0q„) = Q, p W pP'd w) + q0pu), such a chain connectivity T, through which tliei passes a 2perfectly definite integral of the equation F-Utaking 00 initial chain connectivities T, as for instance by taking the curves x0 = 6, y0 = P, z0 = p to2 lie the co2 curves^ upon an arbitrary surface, we thus obtain oo integrals, and so oo e
