450
DIFFERENTIAL EQUATIONS a determinant of r rows and columns not vanishing identically is dx„, y0 + dy„, z0 + S1^), namely, 5lz0 ought to be equal to dh0. But that formed by the coefficients of the differential coefficients of/ we find d1z0=a0dx0 + b(x0 + dx0, ya, z0 + a0dx0)dy0 = a0dx0+b0dy0 regard to x1 • • xr; also that the coefficients are all Complete in +a0Jr ^,and so at once reach the condition of inteabout the values x1=x1° • • xn = xn°, and + dx0dy0 systems developable for these values the determinant just spoken of is of linear that grability. If now we put x = x0 + t, y = y0 + mt, and regard m as not zero. Then the main theorem is that the complete partial s stem constant, we shall fact be considering the section of the surface by ^equations set y of kof equations, r equations,have and therefore the originally given a fixed plane y -yin 0 = m(x-x0); along this section dz=dt(a+bm)in common n-r solutions, say wr+1 • • un, which reduce respectively to xr+1 • . xn when in if we then integrate the equation dx/dt^a + bm, where a, b are’ them for xx ■ • xr are respectively put aq0 • • xr0-, so that also0 the expressed as functions of m and t, with m kept constant, finding equations have in common a solution reducing when aq = aq • • the solution which reduces to z0 for £ = 0, and in the result again xr=xr° to an arbitrary function ^{xr+1 • • xn) which is developable replace m by (y - y0)l(x - x0), we shall have the surface in question. about xr°+1 • • xn°, namely, this common solution is i/4 wr+1 • • w„). In the general case the equations dxj=clidx1 + • • + A,a yer s - ’ It is seen at once that this result is a generalization of the theorem Crjdxr similarly determine through an arbitrary point etbod for r=l, and its proof is conveniently given by induction from aq° • • xn° a planar manifold of r dimensions in space ™,te ramof that case. It can be verified without difficulty (1) that if from of n dimensions, and when the conditions of integra- 'Jtwa% ' the r equations of the complete system we form r independent bility are satisfied, every direction in this manifold linear aggregates, with coefficients not necessarily constants, the through this point is tangent to the manifold of r dimensions, new system is also a complete system ; (2) that if in place of the expressed by Ur+ = xr°+l • • 03n = xn°, which satisfies the equations independent variables aq • • xn we introduce any other variables and passes through this point. If we put aq - x1°=t, x2 - x2°=m2t, which are independent functions of the former, the new equations ■ xr-xr° — mrt, and regard m2- ■ mr as fixed, the (n-r) total also form a complete system. It is convenient, then, from the equations take the form dxjdt=-c-lj + m2c2j+• •+mrcrj, and their complete system of r equations to form r new equations by solving integration is equivalent to that of the single partial equation dfdt + separately for dfdxx, • •, dfdxr; suppose the general equation of S (c + m c +' • + mrcrj)dfldxj — 0 in the n-r+1 variables t, the new system to be Qvf=d/ldx(r + c„r+1dfldxr+1+- ■ +ctrndfldxn J‘r+1 lj 2 2i = 0(<r = l • • r). Then it is easily obvious that the equation QpQjf xr+1 • ■ xn. Determining the solutions Or+1 • • On which reduce - Q<7Qp/= 0 contains only the differential coefficients of/ in regard to respectively xr+1 • • x„ when t = 0, and putting therein t = xl- aq°, to av+i • • Xnl as it is at most a linear function of (//, • •, Q,/, it ^2 (^■2 •^2°)/(^'i Xy’), ■ *, mr—(aq. xr°)j(Xj aq0), we obtain the must be identically zero. So reduced the system is called a Jaco- solutions of the original system of partial equations previously system. Of this system Q1/= 0 has n-1 principal denoted by wr+1 • • w„. It is to be remarked, however, that the Jacobian bian solutions reducing respectively to x2 ■ • xn when aq = presence of the fixed parameters m2 • • mr in the single integration 0 systems. aq , and its form shows that of these the first r — 1 are may frequently render it more difficult than if they were assigned exactly aq • • xr. Let these n-1 functions together with aq be numerical quantities. We have above considered the integration of an equation dz= introduced as n new independent variables in all the r equations. Since the first equation is satisfied by n - 1 of the new independent adx + bdy on the hypothesis that the condition dajdy+ bdajdz= dbjdx + adbjdz. It is natural to inquire what relations variables, it will contain no differential coefficients in regard to faffla them, and will reduce therefore simply to dfldx1 = 0, expressing among x, y, z, if any, are implied by, or are consistent Pbx reSmn with, a differential relation adx + bdy + cdz = o, when a, P that any common solution of the r equations is a function only s oas ’ ■of the n-1 remaining variables. Thereby the investigation of b, c are unrestricted functions of x. y, z. This problem the common solutions is reduced to the same problem for r-1 leads to the consideration of the so-called Pfaffian Expression equations in n -1 variables. Proceeding thus, we reach at length adx + bdy + cdz; it can be shown (1) if each of the quantities one equation in n — r+1 variables, from which, by retracing the dbjdz - dcjdy, dcjdx - da/dz, dajdy - dbjdx, which we shall denote respectively by un, u31, u32, be identically zero, the expression is analysis, the proposition stated is seen to follow. The analogy with the case of one equation is, however, still closer. the differential of a function of x, y, z, equal to dt say ; (2) that if With the coefficients Co-,,- of the equations Q^/=0 in transposed the quantity au23 + bu33 + cu12 is identically zero, the expression is (<r = l • • r, j—r+1 • • n) we can put down the of the form udt, that is can be made a perfect differential by System of array (n-r) equations, dxj—cvdx1 + - • +cvidxr, equivalent to multiplication by the factor -i ; (3) that in general the expression total dif- the equations dxjldxa.=caj. That consistent ferential withr(n-r) them we may be able to regard xr+l • • xn as is of the form dt + uydtj. Consider the matrix of four rows and equations. functions of aq • • xr, these being regarded as independ- three columns, in which the elements of the first row are a, b, c, ent variables, it is clearly necessary that when we differentiate and the elements of the (r+l)-th row, for r=l, 2, 3, are the Crj in regard to aq on this hypothesis the result should be the quantities url, ur2, ur3, where = 0. Then it is easily same as when we differentiate cPj in regard to xa on this hypothesis. seen that the cases (1), (2), (3) above correspond respectively to The differential coefficient of a function / of x1 - • xn on this the cases when (1) every determinant of this matrix of two rows hypothesis, in regard to xp, is, however, dfjdxp + cp,r+1dfldxr+-i + • • and columns is zero, (2) every determinant of three rows and --Cpndfdxn, namely, is Qp/ Thus the consistence of the n-r total columns is zero, (3) when no condition is assumed. This result ■equations requires the conditions Q caj = 0, which are, how- can be generalized as follows: if cq • • an be any functions of aq • • x„, ever, verified in virtue of QpCQo/) - Q/Q/^O. And it can in the so-called Pfaffian expression a1dx1 + • • + andxn can be reduced fact be easily verified that if wr+1 • -oin be the principal solutions to one or other of the two forms uYdtx + • • + ukdtk, dt + u^dt^ + ■ • + •of the Jacobian system, 1^/= 0, reducing respectively to xr+1 • • aqt wherein t, uY, • -, q, • • are independent functions of aq • • aq,, when aq = aq° • • xr=xr°, and the equations wr+1 = xr°+1 • • 0cjn=xn° and k is such that in these two cases respectively 2k or 2k-1 is the be solved for aq.+1 • • aq, to give xi=f/s(x1 • • aq., aq.°+1 • • aq, ), 0 these rank of a certain matrix of n +1 rows and n columns, that is, the values solve the total equations and reduce respectively to aq. +1 • • greatest number of rows and columns in a non-vanishing determin.aq,0 when aq=aq° • • xr=xr°. And the total equations have no ant of the matrix; the matrix is that whose first row is constiother solutions with these initial values. Conversely, the existence tuted by the quantities ax • • an, whose s-th element in the (r +1) -th ■of these solutions of the total equations can be deduced d priori row is the quantity dajdx, - dasjdxr. The proof of such a and the theory of the Jacobian system based upon them. The reduced form can be obtained from the two results : (1) If t be any theory of such total equations, in general, finds its natural place given function of the 2m independent variables ux • • umtx • • tm, Tinder the heading Pfaffian Expressions, below. the expression dt + uxdtx + • • -i-umdtm can be put into the form A practical method of reducing the solution of the r equations udt + • • +u'mdt'm. (2) If the quantities ux • • umtx • ■ tm be of a Jacobian system to that of a single equation in n-r+1 connected by a relation, the expression uxdtx + • • + umdtm can be may be explained in connexion with a geo- put into the form dt' + uxdt'x + • • + um_xdt'm_x; and if the relation Geometri- variablesinterpretation which will perhaps be clearer in connecting ux • ■ umtx • • tmbo homogeneous in ux • • um, then t' can cal inter- metrical particular case, sayw = 3, r = 2. There is then only be taken to be zero. These two results are deductions from the pretation aone total equation, say dz = adx + bdy ; if we do not theory of contact transformations (see below), and their demonstraand account of the condition of integrability, which tion requires, beside elementary algebraical considerations, only the solution. istakehere dajdy + bdaldz=dbldx + adbldz, this equation theory of complete systems of linear homogeneous partial differenmay be regarded as defining through an arbitrary point x0y0z0 tial equations of the first order. When the existence of the reduced of three-dimensioned space (about which a, b are developable) a form of the Pfaffian expression containing only independent quanplane, namely, z-z0 =2a0(x-x0) + b0(y-y0), and therefore, through tities is thus once assured, the identification of the number k with this arbitrary point <x> directions, namely, all those in the plane. that defined by the specified matrix may, with some difficulty, be If now there be a surface z = p(xy), satisfying dz = adx + bdy and made d posteriori. In all cases of a single Pfaffian equation we are thus led to passing through xGy0z0, this plane will touch the surface, and the operations of passing along the surface from (x0yrjz0) to (xo + dx0, consider what is implied by a relation dt-uxdtx- • • -umdtm=t), ya, z„ + dz0) and then to l(x0 + dx0, yo + dy0, z0 + dho), ought to lead in which t,ux • • umtx • • tm are, except for this equation, independto the same value of d z0 as do the operations of passing along ent variables. This is to be satisfied in virtue of one or several the surface from (x0y^0) to (x0) y0 + dy0, z0 + Sz0), and then to (x0 + relations connecting the variables ; these must involve relations con-
