DIFFERENTIAL
EQUATIONS
451
nee ting t, t1- • tm only, and in one of these at least t must actually (non-homogeneous) contact transformation ; (2) the ZXj • • Xn enter. We can then suppose that in one actual system of relations verify the [n +1) identities [ZX,-]=0, [XjXy] = 0. And the in virtue of which the Pfaffian equation is satisfied, all further identities [PiXi] = cr, [PiX,-] = 0, [PiZ] = crPi, [P*P>] = 0, [Z<r] = Single the relations connecting t,t ■ • tm only are given by t = ad— [X,cr] = cr q [Pi(r] = cr are also verified. Conlinear '/'(^s+l ■ ’ in'.-h h — ’/'rC^s+l ■ • ^m) ’ • ts = ps{ts+1 • • tm) ; SO dz dz dz Pfaffian that the equation dp u dp • • u dp u dt 1 1 s s s+1 s+1 versely, if ZX • • X be independent functions satisfying the idenx n equation. • • -u dt = Q is identically true in regard to iq • • u m m m tities [ZXi] = 0, [X£X_,] = 0, then a, other than zero, and Pj • • P„ can equating to zero the coefficients of the differentials of be uniquely determined, by solution of algebraic equations, such «+i t these variables, we thus obtain m-s relations of the form d^jdtj that drL — PjrfXj — • • — PndXn = <r[dz-p1dx1 pndxn). Finally, -u^Jdtj- • • - usd^ps!dtj -Uj — 0 ; these m-s relations, with the there is a particular case of great importance arising when (7=1, previous s + 1 relations, constitute a set of m +1 relations connect- which gives the results: (1) If UXj • • XnPj • • PTC be 2ji + 1 funcing the 2m +1 variables in virtue of which the Pfaffian equation is tions of the 2n independent variables aq • • x p1 • • pn, satisfying satisfied independently of the form of the functions i//, i/q • • {/s. the identity <fU + PjdX, + • • + P a!X =yqdaq + -n - +p dx , then the There is clearly such a set for each of the values s = 0, s = l, • •, 2n functions Pj • • P^Xj • • Xn naren independent, nandn we have s—rn-l, m. And for any value of s there may exist relations (XiXy) = 0, (XjU) = 5Xf, (PfX<) = l, (PiXy)=0, (PiP,) = 0, (PiU) + additional to the specified m + 1 relations, provided they do not P = 5P,-, where S denotes the operator p ddp + • -+p dldpn; (2) If involve any relation connecting t, • - tm only, and are consistent XjX • • Xn be independent functions ofx aq • •Y xnPi • • pn n, such that with the m-s relations connecting u1 ■ • umIt is now evident (XjXy) = 0, thenU can be found by a quadrature, such that (XiIJ) = that, essentially, the integration of a Pfaffian equation a1dx1 + • • SXg; and when X* • • X„, U satisfy these ln[n + l) conditions, then + aHdxn—0, wherein ax • • an are functions of aq • • xn, is effected Pj • • P„ can be found, by solution of linear algebraic equations, to by the processes necessary to bring it to its reduced form, involving render true the identity c£U + P^Xj -I b VndXn=p1dx^ + • • +pndxn; •only independent variables. And it is easy to see that if we (3) Functions Xx • • XMPj • • P„ can be found to satisfy this differsuppose this reduction to be carried out in all possible ways, there ential identity when U is an arbitrary given function of is no need to distinguish the classes of integrals corresponding to aq • • aqpr - ■ pn', but this requires integrations. In order to see the various values of s ; for it can be verified without difficulty what integrations, it is only necessary to verify the statement that that by putting t'—t- ufa =- • • - usts, =t=%,••, t’s=us, = -tx, if U be an arbitrary given function of aq • • aq^q • • pn, and, for r<?i, IL s— ~ts, t s_)_i ‘ ' t m imi 'U «+l ^s+l ’ • u m=um, the reduced Xj • • X be independent functions of these variables, such that equation becomes changed to dt' - udt - • • - u'mdt'm = 0, and the (X^U) =r SXa, (XpXJ = 0, for p, cr = 1 • ■ r, then the r+ homogeneous general relations changed to t' = f/{t‘g+1 ■ • tm) — t linear partial differential equations of the first order (U/)+S/^O, S+1 • • tm) -. . • • t'm), = <(>, say, together with u1 = d(pldt' 1, • •, (Xp/) = 0, form a complete system. It will be seen that the assumpu'm=d<pldt’m, which contain only one relation connecting the vari- tions above made for the reduction of Pfaffian expressions follow ables t' t • • t'm only. from the results here enunciated for contact transformations. This method for a single Pfaffian equation can, strictly speaking, be generalized to a simultaneous system of (n-r) Pfaffian equations We pass on now to consider the solution of any partial dxj — clJdx1 + • • + c-rjdxr only in the case already treated, Simu - when this system is satisfied by regarding xr+1 ■ • xn as differential equation of the first order; we attempt to Pfaffian su'tablecase functions of the independent variables x1 • • xr; explain certain ideas relatively to a single equa- PartiaI le q eauatlons ^ integral manifolds are may of r dimensions. ’ When these are non-existent, there be integral tion with any number of independent variables different manifolds of higher dimensions; for if d<p=(p1dx1+ • • + (f>^dxr + (in particular, an ordinary equation of the first tialequa(pr+iiCir+idxj + • • + Crr+idXr) + <pr+i( ) + • • be identically zero, order with one independent variable) by speaking Hon of the then 00- + cirr+1(/)r+j + • • + c^epn = 0, or 0 satisfies the r partial of a single equation with two independent varidifferential equations previously associated with the total equations; when these are not a complete system, but included in a complete ables x, y, and one dependent variable z. It will .system of r + /U equations, having therefore n-r-p independent be seen that we are naturally led to consider systems of such integrals, the total equations are satisfied over a manifold of r + p simultaneous equations, of which we give some account •dimensions. See E. v. Weber, Math. Annal. Iv. (1901), p. 386. below. The central discovery of the transformation theory It seems desirable to add here certain results, largely of algebraic the solution of an equation F(,r, y, z, dzjdx, dz/dy) = 0 •character, which naturally arise in connexion with the theory of of is that its solution can be reduced to the solution of partial contact transformations. For any two functions of the equations which are linear. For this, however, we must transform *ndePendent variables x1 • • xnp1 ■ • pn we denote by (00) regard dzjdx, dzjdy, during the process of integration, not atlons. the sum of the n terms such as ^ ^ . For dpidxi., dpidXi as the differential coefficients of a function z in regard to two functions of the (2?i + l) independent variables zx1 • • xnpl • ■ pn x and y, but as variables independent of x, y, z, the too we denote by [00] the sum of the n terms such as —f — great indefiniteness that might thus appear to be introdpdxi dz) •d0/d0 dej) ). « can at once be verified that for any three duced being provided for in another way. We notice, in fact, that if z = 0(x,y) be a solution of the differential equa■dpiKdXi^dz functions [/[00]] + [0[0/]] + [0L/0]]=^[00] + g[0/] + g' [/0], tion, then dz — dxdxj/jdx + dydipjdy; thus if we denote the equation by ~F(xyzpq) = 0, and prescribe the condition which when /, 0, 0 do not contain z becomes the identity dz=pdx + qdy for every solution, any solution such as (/(00)) + (0(0/)) + (0(/0)) = O. Then if Xj • • • • P„ be such functions of oq • • xrlpY • - pn that P^X^- • + Pn^X„ is identically z = ij/(x, y) will necessarily be associated with the equations equal to pjdaq + • • +pndxn, it can be shown by elementary algebra, p — dz/dx, q —dzjdy, and z will satisfy the equation in its after equating coefficients of independent differentials, (1) that original form. We have previously seen (under Pfaffian the functions Xj • • • • Pn are independent functions of the 2n vari- Expressions) that if five variables xyzpq, otherwise inables aq • • ■ • pn, so that the equations x’i = X,-, yi=Pf can be solved dependent, be subject to dz -pdx - qdy = 0, they must in for aq • • xnpl • • pnt and represent therefore a transformation, which we call a homogeneous contact transformation ; (2) that the Xr- • Xn fact be subject to at least three mutual relations. If we are homogeneous functions of Pi--pn of zero dimensions, the associate with a point xyz the plane Z - z =p(X -x) + Pi. . Pn are homogeneous functions of px ■ • pn of dimension one, q(Y - y) passing through it, where X, Y, Z are current and the (n-) relations (X<Xy)=0 are verified. So also are the ai2 relations (P^) = 1, (PjX,-) = 0, (P<P^ = 0. Conversely, if Xj • • Xn co-ordinates, and call this association a surface-element; be independent functions, each homogeneous of zero dimension in and if two consecutive elements of which the point Pi • -pn, satisfying the (n -1) relations (XfX,-) = 0, then Px • • Pn (x + dx, y + dy, z + dz) of one lies on the plane of the other, can be uniquely determined, by solving linear algebraic equations, for which, that is, the condition dz =pdx + qdy is satisfied, such that P1<2X1+- • + P,/iX„ =pldxl + • • +pndxn. If now we put be said to be connected, and an infinity of connected «• +1 for n, put 2 for xn+l, Z for X„+1, Qj for - P,-/P„+1, for i=- ■ n, put qi for-pilpn+1 and cr for qn+ilQn+i, and then finally write elements following one another continuously be called is that a connectivity Pi ‘ ‘ P«Pi • ■ pn for Qx • • Qnq1 • . qn, we obtain the following re- a connectivity, then our statement sults : If ZXj • • XnPj • • P„ be functions of zXj • ■ xnp1 ■ • pn, such consists of not more than oo 2 elements, the whole number that the expression d'L - P^Xj - . • - PM^XM is identically equal to of elements {xyzpq) that are possible being called oo5. [dz-pxdxx- • --pndxn), and cr not zero, then (1) the functions ZXj ■ • XJ^ • - P„ are independent functions of zaq • • xnpl • • pn, so The solution of an equation F(aq y, z, dzjdx, dzjdy) = 0 is that the equations z =Z, x't- = X,-, p'i = P, can be solved for then to be understood to mean finding in all possible ways, -2Xi • • tCnPi • • Pm and determine a transformation which we call a from the oo 4 elements (xyzpq) which satisfy Y(xyzpq) = 0
