The solution of this partial equation is thus reduced to the solution of the two ordinary differential equations expressed by dx/P = dy/Q = dz/R. In regard to this problem one remark may be made which is often of use in practice: when one equation u = a has been found to satisfy the differential equations, we may utilize this to obtain the second equation v = b; for instance, we may, by means of u = a, eliminate z—when then from the resulting equations in x and y a relation v = b has been found containing x and y and a, the substitution a = u will give a relation involving x, y, z.
Note on Jacobian Determinants.—The fact assumed above that the vanishing of the Jacobian determinant whose elements are the partial derivatives of three functions F, u, v, of three variables x, y, z,involves that there exists a functional relation connecting the three functions F, u, v, may be proved somewhat roughly as follows:—
The corresponding theorem is true for any number of variables. Consider first the case of two functions p, q, of two variables x, y. The function p, not being constant, must contain one of the variables, say x; we can then suppose x expressed in terms of y and the function p; thus the function q can be expressed in terms of y and the function p, say q = Q(p, y). This is clear enough in the simplest cases which arise, when the functions are rational. Hence we have
these give
by hypothesis ∂p/∂x is not identically zero; therefore if the Jacobian determinant of p and q in regard to x and y is zero identically, so is ∂Q/∂y, or Q does not contain y, so that q is expressible as a function of p only. Conversely, such an expression can be seen at once to make the Jacobian of p and q vanish identically.
Passing now to the case of three variables, suppose that the Jacobian determinant of the three functions F, u, v in regard to x, y, z is identically zero. We prove that if u, v are not themselves functionally connected, F is expressible as a function of u and v. Suppose first that the minors of the elements of ∂F/∂x, ∂F/∂y, ∂F/∂z in the determinant are all identically zero, namely the three determinants such as
then by the case of two variables considered above there exist three functional relations. ψ1(u, v, x) = 0, ψ2(u, v, y) = 0, ψ3(u, v, z) = 0, of which the first, for example, follows from the vanishing of
We cannot assume that x is absent from ψ1, or y from ψ2, or z from ψ3; but conversely we cannot simultaneously have x entering in ψ1, and y in ψ2, and z in ψ3, or else by elimination of u and v from the three equations ψ1= 0, ψ2= 0, ψ3= 0, we should find a necessary relation connecting the three independent quantities x, y, z; which is absurd. Thus when the three minors of ∂F/∂x, ∂F/∂y, ∂F/∂z in the Jacobian determinant are all zero, there exists a functional relation connecting u and v only. Suppose no such relation to exist; we can then suppose, for example, that
is not zero. Then from the equations u(x, y, z) = u, v(x, y, z) = v we can express y and z in terms of u, v, and x (the attempt to do this could only fail by leading to a relation connecting u, v and x, and the existence of such a relation would involve that the determinant
was zero), and so write F in the form F(x, y, z) = Φ(u, v, x). We then have
thereby the Jacobian determinant of F, u, v is reduced to
by hypothesis the second factor of this does not vanish identically; hence ∂Φ/∂x = 0 identically, and Φ does not contain x; so that F is expressible in terms of u, v only; as was to be proved.
Part II.—General Theory.
Differential equations arise in the expression of the relations between quantities by the elimination of details, either unknown or regarded as unessential to the formulation of the relations in question. They give rise, therefore, to the two closely connected problems of determining what arrangement of details is consistent with them, and of developing, apart from these details, the general properties expressed by them. Very roughly, two methods of study can be distinguished, with the names Transformation-theories, Function-theories; the former is concerned with the reduction of the algebraical relations to the fewest and simplest forms, eventually with the hope of obtaining explicit expressions of the dependent variables in terms of the independent variables; the latter is concerned with the determination of the general descriptive relations among the quantities which are involved by the differential equations, with as little use of algebraical calculations as may be possible. Under the former heading we may, with the assumption of a few theorems belonging to the latter, arrange the theory of partial differential equations and Pfaff’s problem, with their geometrical interpretations, as at present developed, and the applications of Lie’s theory of transformation-groups to partial and to ordinary equations; under the latter, the study of linear differential equations in the manner initiated by Riemann, the applications of discontinuous groups, the theory of the singularities of integrals, and the study of potential equations with existence-theorems arising therefrom. In order to be clear we shall enter into some detail in regard to partial differential equations of the first order, both those which are linear in any number of variables and those not linear in two independent variables, and also in regard to the function-theory of linear differential equations of the second order. Space renders impossible anything further than the briefest account of many other matters; in particular, the theories of partial equations of higher than the first order, the function-theory of the singularities of ordinary equations not linear and the applications to differential geometry, are taken account of only in the bibliography. It is believed that on the whole the article will be more useful to the reader than if explanations of method had been further curtailed to include more facts.
When we speak of a function without qualification, it is to be understood that in the immediate neighbourhood of a particular set x0, y0, ... of values of the independent variables x, y, ... of the function, at whatever point of the range of values for x, y, ... under consideration x0, y0, ... may be chosen, the function can be expressed as a series of positive integral powers of the differences x − x0, y − y0, ..., convergent when these are sufficiently small (seeFunction: Functions of Complex Variables). Without this condition, which we express by saying that the function is developable about x0, y0, ..., many results provisionally stated in the transformation theories would be unmeaning or incorrect. If, then, we have a set of k functions, ƒ1... ƒkof n independent variables x1... xn, we say that they are independent when n ≥ k and not every determinant of k rows and columns vanishes of the matrix of k rows and n columns whose r-th row has the constituents dƒr/dx1, ... dƒr/dxn; the justification being in the theorem, which we assume, that if the determinant involving, for instance, the first k columns be not zero for x1= xº1... xn= xºn, and the functions be developable about this point, then from the equations ƒ1= c1, ... ƒk= ckwe can express x1, ... xkby convergent power series in the differences xk+1− xºk+1, ... xn− xnº, and so regard x1, ... xkas functions of the remaining variables. This we often express by saying that the equations ƒ1= c1, ... ƒk= ckcan be solved for x1, ... xk. The explanation is given as a type of explanation often understood in what follows.
We may conveniently begin by stating the theorem: If each of the n functions φ1, ... φnof the (n + 1) variables x1, ... xnt be developableOrdinary equations of the first order.about the values xº1, ... xn0t0, the n differential equations of the form dx1/dt = φ1(tx1, ... xn) are satisfied by convergent power series
xr= xºr+ (t − t0) Ar1+ (t − t0)² Ar2+ ...
reducing respectively to xº1, ... xºnwhen t = t0; and the only functions satisfying the equations and reducing respectively to xº1, ... xºnwhen t = t0, are those determined by continuation of these series. If the result of solving these n equations for xº1, ... xºnbe written in the form ω1(x1, ... xnt) = xº1, ... ωn(x1, ... xnt) = xºn,Single homogeneous partial equation of the first order.it is at once evident that the differential equation
dƒ/dt + φ1dƒ/dx1+ ... + φndƒ/dxn= 0
possesses n integrals, namely, the functions ω1, ... ωn, which are developable about the values (xº1... xn0t0) and reduce respectively to x1, ... xnwhen t = t0. And in fact it has no other integrals so reducing. Thus this equation also possesses a unique integral reducing when t = t0to an arbitrary function ψ(x1, ... xn), this integral being. ψ(ω1, ... ωn). Conversely the existence of theseprincipalintegrals ω1, ... ωnof the partial equation establishes the existence of the specified solutions of the ordinary equations dxi/dt = φi. The following sketch of the proof of the existence of these principal integrals for the case n = 2 will show the character of more general investigations. Put x for x − x0, &c., and consider the equation a(xyt) dƒ/dx + b(xyt) dƒ/dy = dƒ/dt, wherein the functions a, b are developable about x = 0, y = 0, t = 0; say
a(xyt) = a0+ ta1+ t²a2/2! + ..., b(xyt) = b0+ tb1+ t²b2/2! + ...,
so that
ad/dx + bd/dy = δ0+ tδ1+ ½t²δ2+ ...,
where δ = ard/dx + brd/dy. In order that
ƒ = p0+ tp1+ t²p2/2! + ...
wherein p0, p1... are power series in x, y, should satisfy the equation, it is necessary, as we find by equating like terms, that
p1= δ0p0, p2= δ0p1+ δ1p0, &c.
and in generalProof of the existence of integrals.
ps+1= δ0ps+ s1δ1ps-1+ s2δ2ps-2+... + δsp0,
where
sr= (s!)/(r!) (s − r)!
Now compare with the given equation another equation
A(xyt)dF/dx + B(xyt)dF/dy = dF/dt,
wherein each coefficient in the expansion of either A or B is real and positive, and not less than the absolute value of the corresponding coefficient in the expansion of a or b. In the second equation let us substitute a series
F = P0+ tP1+ t²P2/2! + ...,
wherein the coefficients in P0are real and positive, and each not less than the absolute value of the corresponding coefficient in p0; then putting Δr= Ard/dx + Brd/dy we obtain necessary equations of the same form as before, namely,
P1= Δ0P0, P2= Δ0P1+ Δ1P0, ...
and in general Ps+1= Δ0Ps, + s1Δ1Ps-1+ ... + ΔsP0. These give for every coefficient in Ps+1an integral aggregate with real positive coefficients of the coefficients in Ps, Ps-1, ..., P0and the coefficients in A and B; and they are the same aggregates as would be given by the previously obtained equations for the corresponding coefficients in ps+1in terms of the coefficients in ps, ps-1, ..., p0and the coefficients in a and b. Hence as the coefficients in P0and also in A, B are real and positive, it follows that the values obtained in succession for the coefficients in P1, P2, ... are real and positive; and further, taking account of the fact that the absolute value of a sum of terms is not greater than the sum of the absolute values of the terms, it follows, for each value of s, that every coefficient in ps+1is, in absolute value, not greater than the corresponding coefficient in Ps+1. Thus if the series for F be convergent, the series for ƒ will also be; and we are thus reduced to (1), specifying functions A, B with real positive coefficients, each in absolute value not less than the corresponding coefficient in a, b; (2) proving that the equation
AdF/dx + BdF/dy = dF/dt
possesses an integral P0+ tP1+ t²P2/2! + ... in which the coefficients in P0are real and positive, and each not less than the absolute value of the corresponding coefficient in p0. If a, b be developable for x, y both in absolute value less than r and for t less in absolute value than R, and for such values a, b be both less in absolute value than the real positive constant M, it is not difficult to verify that we may take A = B = M[1 − (x + y)/r]-1(1 − t/R)-1, and obtain
and that this solves the problem when x, y, t are sufficiently small for the two cases p0= x, p0= y. One obvious application of the general theorem is to the proof of the existence of an integral of an ordinary linear differential equation given by the n equations dy/dx = y1, dy1/dx = y2, ...,
dyn-1/dx = p − p1yn-1− ... − pny;
but in fact any simultaneous system of ordinary equations is reducible to a system of the form
dxi/dt = φi(tx1, ... xn).
Suppose we have k homogeneous linear partial equations of the first order in n independent variables, the general equation being aσ1dƒ/dx1+ ... + aσndƒ/dxn= 0, where σ = 1, ... k, and thatSimultaneous linear partial equations.we desire to know whether the equations have common solutions, and if so, how many. It is to be understood that the equations are linearly independent, which implies that k ≤ n and not every determinant of k rows and columns is identically zero in the matrix in which the i-th element of the σ-th row is aσi}(i = 1, ... n, σ = 1, ... k). Denoting the left side of the σ-th equation by Pσƒ, it is clear that every common solution of the two equations Pσƒ = 0, Pρƒ = 0, is also a solution of the equation Pρ(pσƒ), Pσ(pρƒ), We immediately find, however, that this is also a linear equation, namely, ΣHidƒ/dxi= 0 where Hi= Pρaσ− Pσaρ, and if it be not already contained among the given equations, or be linearly deducible from them, it may be added to them, as not introducing any additional limitation of the possibility of their having common solutions. Proceeding thus with every pair of the original equations, and then with every pair of the possibly augmented system so obtained, and so on continually, we shall arrive at a system of equations, linearly independent of each other and therefore not more than n in number, such that the combination, in the way described, of every pair of them, leads to an equation which is linearly deducible from them. If the number of this so-calledcomplete systemis n, the equations give dƒ/dx1= 0 ... dƒ/dxn= 0, leading to the nugatory result ƒ = a constant. Suppose, then, the number of this system to be r < n; suppose, further, that from theComplete systems of linear partial equations.matrix of the coefficients a determinant of r rows and columns not vanishing identically is that formed by the coefficients of the differential coefficients of ƒ in regard to x1... xr; also that the coefficients are all developable about the values x1= xº1, ... xn= xºn, and that for these values the determinant just spoken of is not zero. Then the main theorem is that the complete system of r equations, and therefore the originally given set of k equations, have in common n − r solutions, say ωr+1, ... ωn, which reduce respectively to xr+1, ... xnwhen in them for x1, ... xrare respectively put xº1, ... xºr; so that also the equations have in common a solution reducing when x1= xº1, ... xr= xºrto an arbitrary function ψ(xr+1, ... xn) which is developable about xºr+1, ... xºn, namely, this common solution is ψ(ωr+1, ... ωn). It is seen at once that this result is a generalization of the theorem for r = 1, and its proof is conveniently given by induction from that case. It can be verified without difficulty (1) that if from the r equations of the complete system we form r independent linear aggregates, with coefficients not necessarily constants, the new system is also a complete system; (2) that if in place of the independent variables x1, ... xnwe introduce any other variables which are independent functions of the former, the new equations also form a complete system. It is convenient, then, from the complete system of r equations to form r new equations by solving separately for dƒ/dx1, ..., dƒ/dxr; suppose the general equation of the new system to be
Qσƒ = dƒ/dxσ+ cσjr+1dƒ/dxr+1+ ... + cσndƒ/dxn= 0 (σ = 1, ... r).
Then it is easily obvious that the equation QρQσƒ − QσQρƒ = 0 contains only the differential coefficients of ƒ in regard to xr+1... xn; as it is at most a linear function of Q1ƒ, ... Qrƒ, it must be identically zero. So reduced the system is called a Jacobian system. Of this system Q1ƒ=0 has n − 1 principal solutions reducing respectivelyJacobian systems.to x2, ... xnwhen
x1= xº1,
and its form shows that of these the first r − 1 are exactly x2... xr. Let these n − 1 functions together with x1be introduced as n new independent variables in all the r equations. Since the first equation is satisfied by n − 1 of the new independent variables, it will contain no differential coefficients in regard to them, and will reduce therefore simply to dƒ/dx1= 0, expressing that any common solution of the r equations is a function only of the n − 1 remaining variables. Thereby the investigation of the common solutions is reduced to the same problem for r − 1 equations in n − 1 variables. Proceeding thus, we reach at length one equation in n − r + 1 variables, from which, by retracing the analysis, the proposition stated is seen to follow.
The analogy with the case of one equation is, however, still closer. With the coefficients cσj, of the equations Qσƒ = 0 in transposed array (σ = 1, ... r, j = r + 1, ... n) we can put down the (n − r) equations, dxj= c1jdx1+ ... + crjdxr, equivalent toSystem of total differential equations.the r(n − r) equations dxj/dxσ= cσr. That consistent with them we may be able to regard xr+1, ... xnas functions of x1, ... xr, these being regarded as independent variables, it is clearly necessary that when we differentiate cσjin regard to xρon this hypothesis the result should be the same as when we differentiate cρj, in regard to xσon this hypothesis. The differential coefficient of a function ƒ of x1, ... xnon this hypothesis, in regard to xρjis, however,
dƒ/dxρ+ cρjr+1dƒ/dxr+1+ ... + cρndƒ/dxn,
namely, is Qρƒ. Thus the consistence of the n − r total equations requires the conditions Qρcσj− Qσcρj= 0, which are, however, verified in virtue of Qρ(Qσƒ) − Qσ(Qρƒ) = 0. And it can in fact be easily verified that if ωr+1, ... ωnbe the principal solutions of the Jacobian system, Qσƒ = 0, reducing respectively to xr+1, ... xnwhen x1= xº1, ... xr= xºr, and the equations ωr+1= x0r+1, ... ωn= xºnbe solved for xr+1, ... xnto give xj= ψj(x1, ... xr, x0r+1, ... xºn), these values solve the total equations and reduce respectively to x0r+1, ... xºnwhen x1= xº1... xr= xºr. And the total equations have no other solutions with these initial values. Conversely, the existence of these solutions of the total equations can be deduced a priori and the theory of the Jacobian system based upon them. The theory of such total equations, in general, finds its natural place under the headingPfaffian Expressions, below.
A practical method of reducing the solution of the r equations of a Jacobian system to that of a single equation in n − r + 1 variables may be explained in connexion with a geometrical interpretation which will perhaps be clearer in a particularGeometrical interpretation and solution.case, say n = 3, r = 2. There is then only one total equation, say dz = adz + bdy; if we do not take account of the condition of integrability, which is in this case da/dy + bda/dz = db/dx + adb/dz, this equation may be regarded as defining through an arbitrary point (x0, y0, z0) of three-dimensioned space (about which a, b are developable) a plane, namely, z − z0= a0(x − x0) + b0(y − y0), and therefore, through this arbitrary point ∞2directions, namely, all those in the plane. If now there be a surface z = ψ(x, y), satisfying dz = adz + bdy and passing through (x0, y0, z0), this plane will touch the surface, and the operations of passing along the surface from (x0, y0, z0) to
(x0+ dx0, y0, z0+ dz0)
and then to (x0+ dx0, y0+ dy0, Z0+ d1z0), ought to lead to the same value of d1z0as do the operations of passing along the surface from (x0, y0, z0) to (x0, y0+ dy0, z0+ δz0), and then to
(x0+ dx0, y0+ dy0, z0+ δ1z0),
namely, δ1z0ought to be equal to d1z0. But we find
and so at once reach the condition of integrability. If now we putx = x0+ t, y = y0+ mt, and regard m as constant, we shall in fact be considering the section of the surface by a fixed plane y − y0= m(x − x0); along this section dz = dt(a + bm); if we then integrate the equation dx/dt = a + bm, where a, b are expressed as functions of m and t, with m kept constant, finding the solution which reduces to z0for t = 0, and in the result again replace m by (y − y0)/(x − x0), we shall have the surface in question. In the general case the equations
dxj= cijdx1+ ... crjdxr
similarly determine through an arbitrary point xº1, ... xºnMayer’s method of integration.a planar manifold of r dimensions in space of n dimensions, and when the conditions of integrability are satisfied, every direction in this manifold through this point is tangent to the manifold of r dimensions, expressed by ωr+1= x0r+1, ... ω_ = xºn, which satisfies the equations and passes through this point. If we put x1− xº1= t, x2− xº2= m2t, ... xr− xºr= mrt, and regard m2, ... mras fixed, the (n − r) total equations take the form dxj/dt = c1j+ m2c2j+ ... + mrcrj, and their integration is equivalent to that of the single partial equation
in the n − r + 1 variables t, xr+1, ... xn. Determining the solutions Ωr+1, ... Ωnwhich reduce to respectively xr+1, ... xnwhen t = 0, and substituting t = x1− xº1, m2= (x2− xº2)/(x1− xº1), ... mr= (xr− xºr)/(x1− xº1), we obtain the solutions of the original system of partial equations previously denoted by ωr+1, ... ωn. It is to be remarked, however, that the presence of the fixed parameters m2, ... mrin the single integration may frequently render it more difficult than if they were assigned numerical quantities.
We have above considered the integration of an equation
dz = adz + bdy
on the hypothesis that the condition
da/dy + bda/dz = db/dz + adb/dz.
It is natural to inquire what relations among x, y, z, if any,Pfaffian Expressions.are implied by, or are consistent with, a differential relation adx + bdy + cdx = 0, when a, b, c are unrestricted functions of x, y, z. This problem leads to the consideration of the so-calledPfaffian Expressionadx + bdy + cdz. It can be shown (1) if each of the quantities db/dz − dc/dy, dc/dx − da/dz, da/dy − db/dz, which we shall denote respectively by u23, u31, u12, be identically zero, the expression is the differential of a function of x, y, z, equal to dt say; (2) that if the quantity au23+ bu31+ cu12is identically zero, the expression is of the form udt,i.e.it can be made a perfect differential by multiplication by the factor 1/u; (3) that in general the expression is of the form dt + u1dt1. Consider the matrix of four rows and three columns, in which the elements of the first row are a, b, c, and the elements of the (r + 1)-th row, for r = 1, 2, 3, are the quantities ur1, ur2, ur3, where u11= u22= u33= 0. Then it is easily seen that the cases (1), (2), (3) above correspond respectively to the cases when (1) every determinant of this matrix of two rows and columns is zero, (2) every determinant of three rows and columns is zero, (3) when no condition is assumed. This result can be generalized as follows: if a1, ... anbe any functions of x1, ... xn, the so-called Pfaffian expression a1dx1+ ... + andxncan be reduced to one or other of the two forms
u1dt1+ ... + ukdtk, dt + u1dt1+ ... + uk-1dtk-1,
wherein t, u1..., t1, ... are independent functions of x1, ... xn, and k is such that in these two cases respectively 2k or 2k − 1 is the rank of a certain matrix of n + 1 rows and n columns, that is, the greatest number of rows and columns in a non-vanishing determinant of the matrix; the matrix is that whose first row is constituted by the quantities a1, ... an, whose s-th element in the (r + 1)-th row is the quantity dar/dxs− das/dxr. The proof of such a reduced form can be obtained from the two results: (1) If t be any given function of the 2m independent variables u1, ... um, t1, ... tm, the expression dt + u1dt1+ ... + umdtmcan be put into the form u′1dt′1+ ... + u′mdt′m. (2) If the quantities u1, ..., u1, t1, ... tmbe connected by a relation, the expression n1dt1+ ... + umdtmcan be put into the format dt′ + u′1dt′1+ ... + u′m-1dt′m-1; and if the relation connecting u1, um, t1, ... tmbe homogeneous in u1, ... um, then t′ can be taken to be zero. These two results are deductions from the theory ofcontact transformations(see below), and their demonstration requires, beside elementary algebraical considerations, only the theory of complete systems of linear homogeneous partial differential equations of the first order. When the existence of the reduced form of the Pfaffian expression containing only independent quantities is thus once assured, the identification of the number k with that defined by the specified matrix may, with some difficulty, be madea posteriori.
In all cases of a single Pfaffian equation we are thus led to consider what is implied by a relation dt − u1dt1− ... − umdtm= 0, in which t, u1, ... um, t1..., tmare, except for this equation, independent variables. This is to be satisfied in virtue ofSingle linear Pfaffian equation.one or several relations connecting the variables; these must involve relations connecting t, t1, ... tmonly, and in one of these at least t must actually enter. We can then suppose that in one actual system of relations in virtue of which the Pfaffian equation is satisfied, all the relations connecting t, t1... tmonly are given by
t = ψ(ts+1... tm), t1= ψ1(ts+1... tm), ... ts= ψs(ts+1... tm);
so that the equation
dψ − u1dψ1− ... − usdψs− us+1dts+1− ... − umdtm= 0
is identically true in regard to u1, ... um, ts+1..., tm; equating to zero the coefficients of the differentials of these variables, we thus obtain m − s relations of the form
dψ/dtj− u1dψ1/dtj− ... − usdψs/dtj− uj= 0;
these m − s relations, with the previous s + 1 relations, constitute a set of m + 1 relations connecting the 2m + 1 variables in virtue of which the Pfaffian equation is satisfied independently of the form of the functions ψ,ψ1, ... ψs. There is clearly such a set for each of the values s = 0, s = 1, ..., s = m − 1, s = m. And for any value of s there may exist relations additional to the specified m + 1 relations, provided they do not involve any relation connecting t, t1, ... tmonly, and are consistent with the m − s relations connecting u1, ... um. It is now evident that, essentially, the integration of a Pfaffian equation
a1dx1+ ... + andxn= 0,
wherein a1, ... anare functions of x1, ... xn, is effected by the processes necessary to bring it to its reduced form, involving only independent variables. And it is easy to see that if we suppose this reduction to be carried out in all possible ways, there is no need to distinguish the classes of integrals corresponding to the various values of s; for it can be verified without difficulty that by putting t′ = t − u1t1− ... − usts, t′1= u1, ... t′s= us, u′1= −t1, ..., u′s= −ts, t′s+1= ts+1, ... t′m= tm, u′s+1= us+1, ... u′m= um, the reduced equation becomes changed to dt′ − u′1dt′1− ... − u′mdt′m= 0, and the general relations changed to
t′ = ψ(t′s+1, ... t′m) − t′1ψ1(t′s+1, ... t′m) − ... − t′sψs(t′s+1, ... t′m), = φ,
say, together with u′1= dφ/dt′1, ..., u′m= dφ/dt′m, which contain only one relation connecting the variables t′, t′1, ... t′monly.
This method for a single Pfaffian equation can, strictly speaking, be generalized to a simultaneous system of (n − r) Pfaffian equations dxj= c1jdx1+ ... + crjdxronly in the case already treated,Simultaneous Pfaffian equations.when this system is satisfied by regarding xr+1, ... xnas suitable functions of the independent variables x1, ... xr; in that case the integral manifolds are of r dimensions. When these are non-existent, there may be integral manifolds of higher dimensions; for if
dφ = φ1dx1+ ... + φrdxr+ φr+1(c1,r+1dx1+ ... + cr,r+1dxr) + φr+2( ) + ...
be identically zero, then φσ + cσ,r+1φr+1+ ... + cσ,nφn≈ 0, or φ satisfies the r partial differential equations previously associated with the total equations; when these are not a complete system, but included in a complete system of r − μ equations, having therefore n − r − μ independent integrals, the total equations are satisfied over a manifold of r + μ dimensions (see E. v. Weber,Math. Annal.1v. (1901), p. 386).
It seems desirable to add here certain results, largely of algebraic character, which naturally arise in connexion with the theory of contact transformations. For any two functions of the 2nContact transformations.independent variables x1, ... xn, p1, ... pnwe denote by (φψ) the sum of the n terms such as dφdψ/dpidxi− dψdφ/dpidxiFor two functions of the (2n + 1) independent variables z, x1, ... xn, p1, ... pnwe denote by φψ the sum of the n terms such as
It can at once be verified that for any three functions [ƒ[φψ]] + [φ[ψƒ]] + [psi[ƒφ]] = dƒ/dz [φψ] + dφ/dz [ψƒ] + dψ/dz [ƒφ], which when ƒ, φ,ψ do not contain z becomes the identity (ƒ(φψ)) + (phi(ψƒ)) + (ψ(ƒφ)) = 0.Then, if X1, ... Xn, P1, ... Pnbe such functions Of x1, ... xn, p1... pnthat P1dX1+ ... + PndXnis identically equal to p1dx1+ ... + pndxn, it can be shown by elementary algebra, after equating coefficients of independent differentials, (1) that the functions X1, ... Pnare independent functions of the 2n variables x1, ... pn, so that the equations x′i= Xi, p′i= Pican be solved for x1, ... xn, p1, ... pn, and represent therefore a transformation, which we call a homogeneous contact transformation; (2) that the X1, ... Xnare homogeneous functions of p1, ... pnof zero dimensions, the P1, ... Pnare homogeneous functions of p1, ... pnof dimension one, and the ½n(n − 1) relations (XiXj) = 0 are verified. So also are the n² relations (PiXi= 1, (PiXj) = 0, (PiPj) = 0. Conversely, if X1, ... Xnbe independent functions, each homogeneous of zero dimension in p1, ... pnsatisfying the ½n(n − 1) relations (XiXj) = 0, then P1, ... Pncan be uniquely determined, by solving linear algebraic equations, such that P1dX1+ ... + PndXn= p1dx1+ ... + pndxn. If now we put n + 1 for n, put z for xn+1, Z for Xn+1, Qifor -Pi/Pn+1, for i = 1, ... n, put qifor -pi/pn+1and σ for qn+1/Qn+1, and then finally write P1, ... Pn, p1, ... pnfor Q1, ... Qn, q1, ... qn, we obtain the following results: If ZX1... Xn, P1, ... Pnbe functions of z, x1, ... xn, p1, ... pn, such that the expression dZ − P1dX1− ... − PndXnis identically equal to σ(dz − p1dx1− ... − pndxn), and σ not zero, then (1) the functions Z, X1, ... Xn, P1, ... Pnare independent functions of z, x1, ... xn, p1, ... pn, so that the equations z′ = Z, x′i= Xi, p′i= Pican be solved for z, x1, ... xn, p1, ... pnand determine a transformation which we call a (non-homogeneous) contact transformation; (2) the Z, X1, ... Xnverify the ½n(n + 1)identities [ZXi] = 0, [XiXj] = 0. And the further identities
[PiXi] = σ, [PiXj] = 0, [PiZ] = σPi, [PiPj] = 0,
are also verified. Conversely, if Z, x1, ... Xnbe independent functions satisfying the identities [ZXi] = 0, [XiXj] = 0, then σ, other than zero, and P1, ... Pncan be uniquely determined, by solution of algebraic equations, such that
dZ − P1dX1− ... − PndXn= σ(dz − p1dx1− ... − pndxn).
Finally, there is a particular case of great importance arising when σ = 1, which gives the results: (1) If U, X1, ... Xn, P1, ... Pnbe 2n + 1 functions of the 2n independent variables x1, ... xn, p1, ... pn, satisfying the identity
dU + P1dx1+ ... + PndXn= p1dx1+ ... + pndxn,
then the 2n functions P1, ... Pn, X1, ... Xnare independent, and we have
(XiXj) = 0, (XiU) = δXi, (PiXi) = 1, (PiXj) = 0, (PiPj) = 0, (PiU) + Pi= δPi,
where δ denotes the operator p1d/dp1+ ... + pnd/dpn; (2) If X1, ... Xnbe independent functions of x1, ... xn, p1, ... pn, such that (XiXj) = 0, then U can be found by a quadrature, such that
(XiU) = δXi;
and when Xi, ... Xn, U satisfy these ½n(n + 1) conditions, then P1, ... Pncan be found, by solution of linear algebraic equations, to render true the identity dU + P1dX1+ ... + PndXn= p1dx1+ ... + pndxn; (3) Functions X1, ... Xn, P1, ... Pncan be found to satisfy this differential identity when U is an arbitrary given function of x1, ... xn, p1, ... pn; 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, ... xn, p1, ... pn, and, for r < n, X1, ... Xrbe 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, anPartial differential equation of the first order.ordinary equation of the first order with one independent 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 arelinear. 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 (underPfaffian 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 beconnected,and an infinity of connected elements following one another continuously be called aconnectivity, 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 ∞4elements (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 equationMeaning 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, ... xn, z, p1, ... pn) = 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 underPfaffian 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 carryingOrder of the ideas.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 x′, y′, z′, p′, q′ be introduced, as before explained underPfaffian Expressions,Characteristic chains.in which z’ is of the form
z′ = z − p1x1− ... − psxs(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
Passing in this direction on the surface z′ = ψ(x′, y′) the tangentplane of the surface at this consecutive point is (p′ + dp′, q′ + dq′), where, since F′(x′, y′, ψ, dψ/dx′, dψ/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,
are satisfied along a connectivity of ∞¹ elements consisting of a curve on z′ = ψ(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′ + δx′, y′+δy′, z′ + δz′) be a consecutive point of this curve, we find at once
thus the equations above give δx′dp′ + δy′dq′ = 0, or the tangent line of the plane curve, is, on the surface z′ = ψ(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 acharacteristic 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 ∞4elements 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 asComplete integral constructed with characteristic chains.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 isnota 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