VOLUME VIII SLICE IV Diameter to Dinarchus (continued 7)

Σun = Σun-1 + ΔΣun-1 = C + un-2 + un-1,

Σun+1 = C + un-2 + un-1 + un,

and so on. This is true whatever C may be, so that the knowledge of ... un-1, un, ... gives us no knowledge of the exact value of Σun; in other words, C is an arbitrary constant, the value of which must be supposed to be the same throughout any operations in which we are concerned with values of Σu corresponding to different suffixes.

There is another symbol E, used in conjunction with u to denote the next term in the series. Thus Eun means un+1, so that Eun = un + Δun.

10. Corresponding to the advancing-difference notation there is a receding-difference notation, in which un+1 − un is regarded as a difference of un+1, and may be denoted by Δ′un+1, and similarly un+1 − 2un + un-1 may be denoted by Δ′²un+1. This notation is only required for certain special purposes, and the usage is not settled (§ 19 (ii.)).

11. The central-difference notation depends on treating un+1 − 2un − un-1 as the second difference of un, and therefore as corresponding to the value xn; but there is no settled system of notation. The following seems to be the most convenient. Since un is a function of xn, and the second difference un+2 − 2un+1 + un is a function of xn+1, the first difference un+1 − un must be regarded as a function of xn+1/2, i.e. of ½(xn + xn+1). We therefore write un+1 − un = δun+1/2, and each difference in the table in § 9 will have the same suffix as the value of x in the same horizontal line; or, if the difference is of an odd order, its suffix will be the means of those of the two nearest values of x. This is shown in the table below.

In this notation, instead of using the symbol E, we use a symbol μ to denote the mean of two consecutive values of u, or of two consecutive differences of the same order, the suffixes being assigned on the same principle as in the case of the differences. Thus

μun+1/2 = ½(un + un+1, μδun = ½(δun-1/2 + δun+1/2, &c.

If we take the means of the differences of odd order immediately above and below the horizontal line through any value of x, these means, with the differences of even order in that line, constitute the central differences of the corresponding value of u. Thus the table of central differences is as follows, the values obtained as means being placed in brackets to distinguish them from the actual differences:—

x u 1st Diff. 2nd Diff. 3rd Diff. 4th Diff.

· · · · · ·

· · · · · ·

· · · · · ·

xn-2 un-2 (μδun-2) δ²un-2 (μδ³un-2) δ4un-2 ...

δun-3/2 δ³un-3/2

xn-1 un-1 (μδun-1) δ²un-1 (μδ³un-1) δ4un-1 ...

δun-1/2 δ³un-2

xn un (μδun) δ²un (μδ³un) δ4un ...

δun+1/2 δ³un+1/2

xn+1 un+1 (μδun+1) δ²un+1 (μδ³un+1) δ4un+1 ...

δun+3/2 δ³un+3/2

xn+2 un+2 (μδun+2) δ²un+2 (μδ³un+2) δ4un+2 ...

· · · · · ·

· · · · · ·

· · · · · ·

Similarly, by taking the means of consecutive values of u and also of consecutive differences of even order, we should get a series of terms and differences central to the intervals xn-2 to xn-1, xn-1 to xn, ....

The terms of the series of which the values of u are the first differences are denoted by σu, with suffixes on the same principle; the suffixes being chosen so that δσun shall be equal to un. Thus, if

σun-3/2 = C + un-2,

then

σun-1/2 = C + un-2 + un-1, σn+1/2 = C + un-2 + un-1 + un, &c.,

and also

μσun-1 = C + un-2 + ½un-1, μσun = C + un-2 + un-1 + ½un, &c.,

C being an arbitrary constant which must remain the same throughout any series of operations.

Operators and Symbolic Methods.

12. There are two further stages in the use of the symbols Δ, Σ, δ, σ, &c., which are not essential for elementary treatment but lead to powerful methods of deduction.

(i.) Instead of treating Δu as a function of x, so that Δun means (Δu)n, we may regard Δ as denoting an operation performed on u, and take Δun as meaning Δ.un. This applies to the other symbols E, δ, &c., whether taken simply or in combination. Thus ΔEun means that we first replace un by un+1, and then replace this by un+2 − un+1.

(ii.) The operations Δ, E, δ, and μ, whether performed separately or in combination, or in combination also with numerical multipliers and with the operation of differentiation denoted by D (≡ d/dx), follow the ordinary rules of algebra: e.g. Δ(un + vn) = Δun + Δvn, ΔDun = DΔun, &c. Hence the symbols can be separated from the functions on which the operations are performed, and treated as if they were algebraical quantities. For instance, we have

E·un = un+1 = un + Δun = 1·un + Δ·un,

so that we may write E = 1 + Δ, or Δ = E − 1. The first of these is nothing more than a statement, in concise form, that if we take two quantities, subtract the first from the second, and add the result to the first, we get the second. This seems almost a truism. But, if we deduce En = (1 + Δ)n, Δn = (E-1)n, and expand by the binomial theorem and then operate on u0, we get the general formulae

un = u0 + nΔu0 + n·n − 1 Δ2u0 + ... + Δnu0,

1·2

Δnu0 = un − nun-1 + n·n − 1 un-2 + ... + (-1)nu0,

1·2

which are identical with the formulae in (ii.) and (i.) of § 3.

(iii.) What has been said under (ii.) applies, with certain reservations, to the operations Σ and σ, and to the operation which represents integration. The latter is sometimes denoted by D-1; and, since ΔΣun = un, and δσun = un, we might similarly replace Σ and σ by Δ-1 and δ-1. These symbols can be combined with Δ, E, &c. according to the ordinary laws of algebra, provided that proper account is taken of the arbitrary constants introduced by the operations D-1, Δ-1, δ-1.

Applications to Algebraical Series.

13. Summation of Series.—If ur, denotes the (r + 1)th term of a series, and if vr is a function of r such that Δvr = ur for all integral values of r, then the sum of the terms um, um+1, ... un is vn+1 − vm. Thus the sum of a number of terms of a series may often be found by inspection, in the same kind of way that an integral is found.

14. Rational Integral Functions.—(i.) If ur is a rational integral function of r of degree p, then Δur, is a rational integral function of r of degree p − 1.

(ii.) A particular case is that of a factorial, i.e. a product of the form (r + a + 1) (r + a + 2) ... (r + b), each factor exceeding the preceding factor by 1. We have

Δ · (r + a + 1) (r + a + 2) ... (r + b) = (b − a)·(r + a + 2) ... (r + b),

whence, changing a into a-1,

Σ(r + a + 1) (r + a + 2) ... (r + b) = const. + (r + a)(r + a + 1) ... (r + b)/(b − a + 1).

A similar method can be applied to the series whose (r + 1)th term is of the form 1/(r + a + 1) (r + a + 2) ... (r + b).

(iii.) Any rational integral function can be converted into the sum of a number of factorials; and thus the sum of a series of which such a function is the general term can be found. For example, it may be shown in this way that the sum of the pth powers of the first n natural numbers is a rational integral function of n of degree p + 1, the coefficient of np+1 being 1/(p + 1).

15. Difference-equations.—The summation of the series ... + un+2 + un-1 + un is a solution of the difference-equation Δvn = un+1, which may also be written (E − 1)vn = un+1. This is a simple form of difference-equation. There are several forms which have been investigated; a simple form, more general than the above, is the linear equation with constant coefficients—

vn+m + a1vn+m-1 + a2vn+m-2 + ... + amvn = N,

where a1, a2, ... am are constants, and N is a given function of n. This may be written

(Em + a1Em-1 + ... + am)vn = N

or

(E − p1)(E − p2) ... (E − pm)vn = N.

The solution, if p1, p2, ... pm are all different, is vn = C1p1n + C2p2n + ... + Cmpmn + Vn, where C1, C2 ... are constants, and vn = Vn is any one solution of the equation. The method of finding a value for Vn depends on the form of N. Certain modifications are required when two or more of the p’s are equal.

It should be observed, in all cases of this kind, that, in describing C1, C2 as “constants,” it is meant that the value of any one, as C1, is the same for all values of n occurring in the series. A “constant” may, however, be a periodic function of n.

Applications to Continuous Functions.

16. The cases of greatest practical importance are those in which u is a continuous function of x. The terms u1, u2 ... of the series then represent the successive values of u corresponding to x = x1, x2.... The important applications of the theory in these cases are to (i.) relations between differences and differential coefficients, (ii.) interpolation, or the determination of intermediate values of u, and (iii.) relations between sums and integrals.

17. Starting from any pair of values x0 and u0, we may suppose the interval h from x0 to x1 to be divided into q equal portions. If we suppose the corresponding values of u to be obtained, and their differences taken, the successive advancing differences of u0 being denoted by ∂u0, ∂²u0 ..., we have (§ 3 (ii.))

u1 = u0 + q∂u0 + q·q − 1 ∂²u0 + ....

1·2

When q is made indefinitely great, this (writing ƒ(x) for u) becomes Taylor’s Theorem (Infinitesimal Calculus)

ƒ(x + h) = ƒ(x) + hƒ′(x) + h² ƒ″(x) + ...,

1·2

which, expressed in terms of operators, is

E = 1 + hD + h² D² + h³ D³ + ... = ehD.

1·2 1·2·3

This gives the relation between Δ and D. Also we have

u2 = u0 + 2q∂u0 + 2q·2q − 1 ∂²u0 + ...

1·2

u3 = u0 + 3q∂u0 + 3q·3q − 1 ∂²u0 + ...

1·2

· ·

· ·

· ·

and, if p is any integer,

up/q = u0 + p∂u0 + p·p − 1 ∂²u0 + ....

1·2

From these equations up/q could be expressed in terms of u0, u1, u2, ...; this is a particular case of interpolation (q.v.).

18. Differences and Differential Coefficients.—The various formulae are most quickly obtained by symbolical methods; i.e. by dealing with the operators Δ, E, D, ... as if they were algebraical quantities. Thus the relation E = ehD (§ 17) gives

hD = loge (1 + Δ) = Δ − ½Δ² + 1⁄3Δ³ ...

or

h(du/dx)0 = Δu0 − ½Δ²u0 + 1⁄3Δ³u0 ....

The formulae connecting central differences with differential coefficients are based on the relations μ = cosh ½hD = ½(e1/2hD + e-1/2hD), δ = 2 sinh ½hD − e1/2hD − e-1/2hD, and may be grouped as follows:—

u0 = u0 }

μδu0 = (hD + 1⁄6 h³D³ + 1⁄120 h5D5 + ...)u0

δ²u0 = (h²D² + 1⁄12 h4D4 + 1⁄360 h6D6 + ...)u0

μδ³u0 = (h³D³ + 1⁄4 h5D5 + ...)u0

δ4u0 = (h4D4 + 1⁄6 h6D6 + ...)u0

· · ·

· · ·

· · ·

μu1/2 = (1 + 1⁄8 h²D² + 1⁄384 h4D4 + 1⁄46080 h6D6 + ...)u1/2 }

δu1/2 = (hD + 1⁄24 h³D³ + 1⁄1920 h5D5 + ...)u1/2

μδ²u1/2 = (h²D² + 5⁄24 h4D4 + 91⁄5760 h6D6 + ...)u1/2

δ³u1/2 = (h³D³ + 1⁄8 h5D5 + ...)u1/2

μδ4 u1/2 = (h4D4 + 7⁄24 h6D6 + ...)u1/2

· · ·

· · ·

· · ·

u0 = u0 }

hDu0 = (μδ − 1⁄6 μδ³ + 1⁄30 μδ5 − ...)u0

h²D²u0 = (δ² − 1⁄12 δ4 + 1⁄90 δ6 − ...)u0

h³D³u0 = (μδ³ − 1⁄4 μδ5 + ...)u0

h4D4u0 = (δ4 − 1⁄6 δ6 + ...)u0

· · ·

· · ·

· · ·

u1/2 = (μ − 1⁄8 μδ² + 3⁄128 μδ4 − 5⁄1024 μδ6 + ...)u1/2 }

hDu1/2 = (δ − 1⁄24 δ³ + 3⁄640 δ5 − ...)u1/2

h²D²u1/2 = (μδ² − 5⁄24 μδ4 + 259⁄5760 μδ6 − ...)u1/2

h³D³u1/2 = (δ³ − 1⁄8 δ5 + ...)u1/2

h4D4 u1/2 = (μδ4 − 7⁄24 μδ6 + ...)u1/2

· · ·

· · ·

· · ·

When u is a rational integral function of x, each of the above series is a terminating series. In other cases the series will be an infinite one, and may be divergent; but it may be used for purposes of approximation up to a certain point, and there will be a “remainder,” the limits of whose magnitude will be determinate.

19. Sums and Integrals.—The relation between a sum and an integral is usually expressed by the Euler-Maclaurin formula. The principle of this formula is that, if um and um+1, are ordinates of a curve, distant h from one another, then for a first approximation to the area of the curve between um and um+1 we have ½h(um + um+1), and the difference between this and the true value of the area can be expressed as the difference of two expressions, one of which is a function of xm, and the other is the same function of xm+1. Denoting these by φ(xm) and φ(xm+1), we have

∫ xm+1 udx = ½h(um + um+1) + φ(xm+1) − φ(xm).

xm

Adding a series of similar expressions, we find

∫ xn udx = h{½um + um+1 + um+2 + ... + un-1 + ½un} + φ(xn) − φ(xm).

xm

The function φ(x) can be expressed in terms either of differential coefficients of u or of advancing or central differences; thus there are three formulae.

(i.) The Euler-Maclaurin formula, properly so called, (due independently to Euler and Maclaurin) is

∫ xn udx = h·μσun − 1⁄12 h² dun ½ + 1⁄720 h4 d³un − 1⁄30240 h6 d5un + ... = h·μσun − B1 h2 dun + B2 h4 d3un − B3 h6 d5un + ...

dx dx3 dx5 2! dx 4! dx3 6! dx5

where B1, B2, B3 ... are Bernoulli’s numbers.

(ii.) If we express differential coefficients in terms of advancing differences, we get a theorem which is due to Laplace:—

1/h ∫ xn udx = μσ(un − u0) − 1⁄12(Δun − Δu0) + 1⁄24(Δ²un − Δ²u0) − 19⁄720(Δ³un − Δ³u0) + 3⁄160(Δ4un − Δ4u0) − ...

x0

For practical calculations this may more conveniently be written

1/h ∫ xn udx = μσ(un − u0) + 1⁄12(Δu0 − ½Δ²u0 + 19⁄60Δ³u0 − ...) + 1⁄12(Δ′ un − ½Δ′ ²un + 19⁄60Δ′ ³un − ...),

x0

where accented differences denote that the values of u are read backwards from un; i.e. Δ′un denotes un-1 − un, not (as in § 10) un − un-1.

(iii.) Expressed in terms of central differences this becomes

1/h ∫ xn udx = μσ(un − u0) − 1⁄12μδun + 11⁄720 μδ³un − ... + 1⁄12μδu0 − 11⁄720 μδ³u0 + ... = μ(σ − 1⁄12δ + 11⁄720δ³ − 191⁄60480δ5 + 2497⁄3628800δ7 − ...)(un − u0).

x0

(iv.) There are variants of these formulae, due to taking hum+1/2 as the first approximation to the area of the curve between um and um+1; the formulae involve the sum u1/2 + u3/2 + ... + un-1/2 ≡ σ(un − u0) (see Mensuration).

20. The formulae in the last section can be obtained by symbolical methods from the relation

1/h ∫ udx = 1/h D-1u = 1/hD · u.

Thus for central differences, if we write θ ≡ ½hD, we have μ = cosh θ, δ = 2 sinh θ, σ = δ-1, and the result in (iii.) corresponds to the formula

sinh θ = θ cosh θ/(1 + 1⁄3 sinh² θ − 2⁄3·5 sinh4 θ + 2·4⁄3·5·7 sinh6 θ − ...).

References.—There is no recent English work on the theory of finite differences as a whole. G. Boole’s Finite Differences (1st ed., 1860, 2nd ed., edited by J. F. Moulton, 1872) is a comprehensive treatise, in which symbolical methods are employed very early. A. A. Markoff’s Differenzenrechnung (German trans., 1896) contains general formulae. (Both these works ignore central differences.) Encycl. der math. Wiss. vol. i. pt. 2, pp. 919-935, may also be consulted. An elementary treatment of the subject will be found in many text-books, e.g. G. Chrystal’s Algebra (pt. 2, ch. xxxi.). A. W. Sunderland, Notes on Finite Differences (1885), is intended for actuarial students. Various central-difference formulae with references are given in Proc. Lond. Math. Soc. xxxi. pp. 449-488. For other references see Interpolation.

(W. F. Sh.)

DIFFERENTIAL EQUATION, in mathematics, a relation between one or more functions and their differential coefficients. The subject is treated here in two parts: (1) an elementary introduction dealing with the more commonly recognized types of differential equations which can be solved by rule; and (2) the general theory.

Part I.—Elementary Introduction.

Of equations involving only one independent variable, x (known as ordinary differential equations), and one dependent variable, y, and containing only the first differential coefficient dy/dx (and therefore said to be of the first order), the simplest form is that reducible to the type

dy/dx = ƒ(x)/F(y),

leading to the result ƒF(y)dy − ƒƒ(x)dx = A, where A is an arbitrary constant; this result is said to solve the differential equation, the problem of evaluating the integrals belonging to the integral calculus.

Another simple form is

dy/dx + yP = Q,

where P, Q are functions of x only; this is known as the linear equation, since it contains y and dy/dx only to the first degree. If ƒPdx = u, we clearly have

d (yeu) = eu ( dy + Py ) = euQ,

dx dx

so that y = e-u(ƒeuQdx + A) solves the equation, and is the only possible solution, A being an arbitrary constant. The rule for the solution of the linear equation is thus to multiply the equation by eu, where u = ƒPdx.

A third simple and important form is that denoted by

y = px + ƒ(p),

where p is an abbreviation for dy/dx; this is known as Clairaut’s form. By differentiation in regard to x it gives

p = p + x dp + ƒ′(p) dp ,

dx dx

where

ƒ′(p) = d ƒ(p);

dp

thus, either (i.) dp/dx = 0, that is, p is constant on the curve satisfying the differential equation, which curve is thus any one of the straight lines y = cx = ƒ(c), where c is an arbitrary constant, or else, (ii.) x + ƒ′(p) = 0; if this latter hypothesis be taken, and p be eliminated between x + ƒ′(p) = 0 and y = px + ƒ(p), a relation connecting x and y, not containing an arbitrary constant, will be found, which obviously represents the envelope of the straight lines y = cx + ƒ(c).

In general if a differential equation φ(x, y, dy/dx) = 0 be satisfied by any one of the curves F(x, y, c) = 0, where c is an arbitrary constant, it is clear that the envelope of these curves, when existent, must also satisfy the differential equation; for this equation prescribes a relation connecting only the co-ordinates x, y and the differential coefficient dy/dx, and these three quantities are the same at any point of the envelope for the envelope and for the particular curve of the family which there touches the envelope. The relation expressing the equation of the envelope is called a singular solution of the differential equation, meaning an isolated solution, as not being one of a family of curves depending upon an arbitrary parameter.

An extended form of Clairaut’s equation expressed by

y = xF(p) + ƒ(p)

may be similarly solved by first differentiating in regard to p, when it reduces to a linear equation of which x is the dependent and p the independent variable; from the integral of this linear equation, and the original differential equation, the quantity p is then to be eliminated.

Other types of solvable differential equations of the first order are (1)

M dy/dx = N,

where M, N are homogeneous polynomials in x and y, of the same order; by putting v = y/x and eliminating y, the equation becomes of the first type considered above, in v and x. An equation (aB ≷ bA)

(ax + by + c)dy/dx = Ax + By + C

may be reduced to this rule by first putting x + h, y + k for x and y, and determining h, k so that ah + bk + c = 0, Ah + Bk + C = 0.

(2) An equation in which y does not explicitly occur,

ƒ(x, dy/dx) = 0,

may, theoretically, be reduced to the type dy/dx = F(x); similarly an equation F(y, dy/dx) = 0.

(3) An equation

ƒ(dy/dx, x, y) = 0,

which is an integral polynomial in dy/dx, may, theoretically, be solved for dy/dx, as an algebraic equation; to any root dy/dx = F1(x, y) corresponds, suppose, a solution φ1(x, y, c) = 0, where c is an arbitrary constant; the product equation φ1(x, y, c)φ2(x, y, c) ... = 0, consisting of as many factors as there were values of dy/dx, is effectively as general as if we wrote φ1(x, y, c1)φ2(x, y, c2) ... = 0; for, to evaluate the first form, we must necessarily consider the factors separately, and nothing is then gained by the multiple notation for the various arbitrary constants. The equation φ1(x, y, c)φ2(x, y, c) ... = 0 is thus the solution of the given differential equation.

In all these cases there is, except for cases of singular solutions, one and only one arbitrary constant in the most general solution of the differential equation; that this must necessarily be so we may take as obvious, the differential equation being supposed to arise by elimination of this constant from the equation expressing its solution and the equation obtainable from this by differentiation in regard to x.

A further type of differential equation of the first order, of the form

dy/dx = A + By + Cy²

in which A, B, C are functions of x, will be briefly considered below under differential equations of the second order.

When we pass to ordinary differential equations of the second order, that is, those expressing a relation between x, y, dy/dx and d²y/dx², the number of types for which the solution can be found by a known procedure is very considerably reduced. Consider the general linear equation

d²y + P dy + Qy = R,

dx² dx

where P, Q, R are functions of x only. There is no method always effective; the main general result for such a linear equation is that if any particular function of x, say y1, can be discovered, for which

d²y1 + P dy1 + Qy1 = 0,

dx² dx

then the substitution y = y1η in the original equation, with R on the right side, reduces this to a linear equation of the first order with the dependent variable dη/dx. In fact, if y = y1η we have

dy = y1 dη + η dy1 and d²y = y1 d²η + 2 dy1 dη + η d²y1 ,

dx dx dx dx² dx² dx dx dx²

and thus

d²y + P dy + Qy = y1 d²η + (2 dy1 + Py1) dη + ( d²y1 + P dy1 + Qy1)η;

dx² dx dx² dx dx dx² dx

if then

d²y1 + P dy1 + Qy1 = 0,

dx² dx

and z denote dη/dx, the original differential equation becomes

y1 dz + ( 2 dy1 Py1) z = R.

dx dx

From this equation z can be found by the rule given above for the linear equation of the first order, and will involve one arbitrary constant; thence y = y1 η = y1 ∫ zdx + Ay1, where A is another arbitrary constant, will be the general solution of the original equation, and, as was to be expected, involves two arbitrary constants.

The case of most frequent occurrence is that in which the coefficients P, Q are constants; we consider this case in some detail. If θ be a root of the quadratic equation θ² + θP + Q = 0, it can be at once seen that a particular integral of the differential equation with zero on the right side is y1 = eθx. Supposing first the roots of the quadratic equation to be different, and φ to be the other root, so that φ + θ = -P, the auxiliary differential equation for z, referred to above, becomes dz/dx + (θ − φ)z = Re-θx which leads to ze(θ-φ) = B + ∫ Re-θxdx, where B is an arbitrary constant, and hence to

y = Aeθ x + eθ x ∫ Be(φ-θ) x dx + eθ x ∫ e(φ-θ) x ∫ Re-θ x dxdx,

or say to y = Aeθ x + Ceθ x + U, where A, C are arbitrary constants and U is a function of x, not present at all when R = 0. If the quadratic equation θ² + Pθ + Q = 0 has equal roots, so that 2θ = -P, the auxiliary equation in z becomes dz/dx = Reθ x giving z = B + ∫ Reθ x dx, where B is an arbitrary constant, and hence

y = (A + Bx)eθ x + eθ x ∫ ∫ Re-θ x dxdx,

or, say, y = (A + Bx)eθ x + U, where A, B are arbitrary constants, and U is a function of x not present at all when R = 0. The portion Aeθ x + Beθ x or (A + Bx)eθ x of the solution, which is known as the complementary function, can clearly be written down at once by inspection of the given differential equation. The remaining portion U may, by taking the constants in the complementary function properly, be replaced by any particular solution whatever of the differential equation

d²v + P dy + Qy = R;

dx² dx

for if u be any particular solution, this has a form

u = A0 eθ x + B0 eφ x + U,

or a form

u = (A0 + B0x) eθ x + U;

thus the general solution can be written

(A − A0)eθ x + (B − B0)eθ x + u, or {A − A0 + (B − B0)x} eθ x + u,

where A − A0, B − B0, like A, B, are arbitrary constants.

A similar result holds for a linear differential equation of any order, say

dny + P1 dn-1y + ... + Pny = R,

dxn dxn-1

where P1, P2, ... Pn are constants, and R is a function of x. If we form the algebraic equation θn + P1θn-1 + ... + Pn = 0, and all the roots of this equation be different, say they are θ1, θ2, ... θn, the general solution of the differential equation is

y = A1 eθ1 x + A2 eθ2 x + ... + An eθn x + u,

where A1, A2, ... An are arbitrary constants, and u is any particular solution whatever; but if there be one root θ1 repeated r times, the terms A1 eθ1 x + ... + Ar eθr x must be replaced by (A1 + A2x + ... + Arxr-1)eθ1 x where A1, ... An are arbitrary constants; the remaining terms in the complementary function will similarly need alteration of form if there be other repeated roots.

To complete the solution of the differential equation we need some method of determining a particular integral u; we explain a procedure which is effective for this purpose in the cases in which R is a sum of terms of the form eaxφ(x), where φ(x) is an integral polynomial in x; this includes cases in which R contains terms of the form cos bx·φ(x) or sin bx·φ(x). Denote d/dx by D; it is clear that if u be any function of x, D(eaxu) = eaxDu + aeaxu, or say, D(eaxu) = eax(D + a)u; hence D²(eaxu), i.e. d²/dx² (eaxu), being equal to D(eaxv), where v = (D + a)u, is equal to eax(D + a)v, that is to eax(D + a)²u. In this way we find Dn(eaxu) = eax(D + a)nu, where n is any positive integer. Hence if ψ(D) be any polynomial in D with constant coefficients, ψ(D) (eaxu) = eaxψ(D + a)u. Next, denoting ∫ udx by D-1u, and any solution of the differential equation dz/dx + az = u by z = (d + a)-1u, we have D[eax(D + a)-1u] = D(eaxz) = eax(D + a)z = eaxu, so that we may write D-1(eaxu) = eax(D + a)-1u, where the meaning is that one value of the left side is equal to one value of the right side; from this, the expression D-2(eaxu), which means D-1[D-1(eaxu)], is equal to D-1(eaxz) and hence to eax(D + a)-1z, which we write eax(D + a)-2u; proceeding thus we obtain

D-n(eaxu) = eax(D + a)-nu,

where n is any positive integer, and the meaning, as before, is that one value of the first expression is equal to one value of the second. More generally, if ψ(D) be any polynomial in D with constant coefficients, and we agree to denote by [1/ψ(D)]u any solution z of the differential equation ψ(D)z = u, we have, if v = [1/ψ(D + a)]u, the identity ψ(D)(eaxv) = eaxψ(D + a)v = eaxu, which we write in the form

1 (eaxu) = eax 1 u.

ψ(D) ψ(D + a)

This gives us the first step in the method we are explaining, namely that a solution of the differential equation ψ(D)y = eaxu + ebxv + ... where u, v, ... are any functions of x, is any function denoted by the expression

eax 1 u + ebx 1 v + ....

ψ(D + a) ψ(D + b)

It is now to be shown how to obtain one value of [1/ψ(D + a)]u, when u is a polynomial in x, namely one solution of the differential equation ψ(D + a)z = u. Let the highest power of x entering in u be xm; if t were a variable quantity, the rational fraction in t, 1/ψ(t + a) by first writing it as a sum of partial fractions, or otherwise, could be identically written in the form

Krt-r + Kr-1t-r+1 + ... + K1t-1 + H + H1t + ... + Hmtm + tm+1φ(t)/ψ(t + a),

where φ(t) is a polynomial in t; this shows that there exists an identity of the form

1 = ψ(t + a)(Krt−r + ... + K1t−1 + H + H1t + ... + Hmtm) + φ(t)tm+1,

and hence an identity

u = ψ(D + a) [KrD−r + ... + K1D−1 + H + H1D + ... + HmDm] u + φ(D) Dm+1u;

in this, since u contains no power of x higher than xm, the second term on the right may be omitted. We thus reach the conclusion that a solution of the differential equation ψ(D + a)z = u is given by

z = (KrD−r + ... + K1D−1 + H + H1D + ... + HmDm)u,

of which the operator on the right is obtained simply by expanding 1/ψ(D + a) in ascending powers of D, as if D were a numerical quantity, the expansion being carried as far as the highest power of D which, operating upon u, does not give zero. In this form every term in z is capable of immediate calculation.

Example.—For the equation

d4v + 2 d²y + y = x³ cos x or (D² + 1)²y = x³ cos x,

dx4 dx3

the roots of the associated algebraic equation (θ² + 1)² = 0 are θ = ±i, each repeated; the complementary function is thus

(A + Bx)eix + (C + Dx)e−ix,

where A, B, C, D are arbitrary constants; this is the same as

(H + Kx) cos x + (M + Nx) sin x,

where H, K, M, N are arbitrary constants. To obtain a particular integral we must find a value of (1 + D²)−²x³ cos x; this is the real part of (1 + D²)−² eixx³ and hence of eix [1 + (D + i)²]−² x³

or

eix [2iD(1 + ½iD)]−² x³,

or

−¼eix D−² (1 + iD − ¾D² − ½iD³ + 5⁄16D4 + 3⁄16iD5 ...)x³,

or

−¼eix (1⁄20x5 + ¼ix4 − ¾x³ − 3⁄2 ix² + 15⁄8 x + 9⁄8 i);

the real part of this is

−¼ (1⁄20 x5 − ¾x² + 15⁄8x) cos x + ¼ (¼x4 − 3⁄4x² + 9⁄8) sin x.

This expression added to the complementary function found above gives the complete integral; and no generality is lost by omitting from the particular integral the terms −15⁄32x cos x + 9⁄32 sin x, which are of the types of terms already occurring in the complementary function.

The symbolical method which has been explained has wider applications than that to which we have, for simplicity of explanation, restricted it. For example, if ψ(x) be any function of x, and a1, a2, ... an be different constants, and [(t + a1) (t + a2) ... (t + an)]−1 when expressed in partial fractions be written Σcm(t + am)−1, a particular integral of the differential equation (D + a1)(D + a2) ... (D + an)y = ψ(x) is given by

y = Σcm(D + am)−1 ψ(x) = Σcm (D + am)−1 e−amxeamx ψ(x) = Σcme−amxD−1 (eamxψ(x) ) = Σcme−amx ∫ eamxψ(x)dx.

The particular integral is thus expressed as a sum of n integrals. A linear differential equation of which the left side has the form

xn dny + P1xn−1 dn−1y + ... + Pn−1x dy + Pny,

dxn dxn−1 dx

where P1, ... Pn are constants, can be reduced to the case considered above. Writing x = et we have the identity

xm dmu = θ(θ − 1)(θ − 2) ... (θ − m + 1)u, where θ = d/dt.

dxm

When the linear differential equation, which we take to be of the second order, has variable coefficients, though there is no general rule for obtaining a solution in finite terms, there are some results which it is of advantage to have in mind. We have seen that if one solution of the equation obtained by putting the right side zero, say y1, be known, the equation can be solved. If y2 be another solution of

d²y + P dy + Qy = 0,

dx² dx

there being no relation of the form my1 + ny2 = k, where m, n, k are constants, it is easy to see that

d (y1′y2 − y1y2′) = P(y1′y2 − y1y2′),

dx

so that we have

y1′y2 − y1y2′ = A exp. (∫ Pdx),

where A is a suitably chosen constant, and exp. z denotes ez. In terms of the two solutions y1, y2 of the differential equation having zero on the right side, the general solution of the equation with R = φ(x) on the right side can at once be verified to be Ay1 + By2 + y1u − y2v, where u, v respectively denote the integrals

u = ∫ y2φ(x) (y1′y2 − y2′y1)−1dx, v = ∫ y1φ(x) (y1′y2 − y2′y1)−1dx.

The equation

d²y + P dy + Qy = 0,

dx² dx

by writing y = v exp. (-½ ∫ Pdx), is at once seen to be reduced to d²v/dx² + Iv = 0, where I = Q − ½dP/dx − ¼P². If η = − 1/v dv/dx, the equation d²v/dx² + Iv = 0 becomes dη/dx = I + η², a non-linear equation of the first order.

More generally the equation

dη = A + Bη + Cη²,

dx

where A, B, C are functions of x, is, by the substitution

η = − 1 dy

Cy dx

reduced to the linear equation

d²y − (B + 1 dC ) dy + ACy = 0.

dx² C dx dx

The equation

dη = A + Bη + Cη²,

dx

known as Riccati’s equation, is transformed into an equation of the same form by a substitution of the form η = (aY + b)/(cY + d), where a, b, c, d are any functions of x, and this fact may be utilized to obtain a solution when A, B, C have special forms; in particular if any particular solution of the equation be known, say η0, the substitution η = η0 − 1/Y enables us at once to obtain the general solution; for instance, when

2B = d log ( A ),

dx C

a particular solution is η0 = √(-A/C). This is a case of the remark, often useful in practice, that the linear equation

φ(x) d²y + ½ dφ dy + μy = 0,

dx² dx dx

where μ is a constant, is reducible to a standard form by taking a new independent variable z = ∫ dx[φ(x)]-½.

We pass to other types of equations of which the solution can be obtained by rule. We may have cases in which there are two dependent variables, x and y, and one independent variable t, the differential coefficients dx/dt, dy/dt being given as functions of x, y and t. Of such equations a simple case is expressed by the pair

dx = ax + by + c, dy a′x + b′y + c′,

dt dt

wherein the coefficients a, b, c, a′, b′, c′, are constants. To integrate these, form with the constant λ the differential coefficient of z = x + λy, that is dz/dt = (a + λa′)x + (b + λb′)y + c + λc′, the quantity λ being so chosen that b + λb′ = λ(a + λa′), so that we have dz/dt = (a + λa′)z + c + λc′; this last equation is at once integrable in the form z(a + λa′) + c + λc′ = Ae(a + λa′)t, where A is an arbitrary constant. In general, the condition b + λb′ = λ(a + λa′) is satisfied by two different values of λ, say λ1, λ2; the solutions corresponding to these give the values of x +λ1y and x + λ2y, from which x and y can be found as functions of t, involving two arbitrary constants. If, however, the two roots of the quadratic equation for λ are equal, that is, if (a − b′)² + 4a′b = 0, the method described gives only one equation, expressing x + λy in terms of t; by means of this equation y can be eliminated from dx/dt = ax + by + c, leading to an equation of the form dx/dt = Px + Q + Re(a + λa′)t, where P, Q, R are constants. The integration of this gives x, and thence y can be found.

A similar process is applicable when we have three or more dependent variables whose differential coefficients in regard to the single independent variables are given as linear functions of the dependent variables with constant coefficients.

Another method of solution of the equations

dx/dt = ax + by + c, dy/dt = a′x + b′y + c′,

consists in differentiating the first equation, thereby obtaining

d²x = a dx + b dy ;

dt² dt dx

from the two given equations, by elimination of y, we can express dy/dt as a linear function of x and dx/dt; we can thus form an equation of the shape d²x/dt² = P + Qx + Rdx/dt, where P, Q, R are constants; this can be integrated by methods previously explained, and the integral, involving two arbitrary constants, gives, by the equation dx/dt = ax + by + c, the corresponding value of y. Conversely it should be noticed that any single linear differential equation

d²x = u + vx + w dx ,

dt² dt

where u, v, w are functions of t, by writing y for dx/dt, is equivalent with the two equations dx/dt = y, dy/dt = u + vx + wy. In fact a similar reduction is possible for any system of differential equations with one independent variable.

Equations occur to be integrated of the form

Xdx + Ydy + Zdz = 0,

where X, Y, Z are functions of x, y, z. We consider only the case in which there exists an equation φ(x, y, z) = C whose differential

∂φ dx + ∂φ dy + ∂φ dz = 0

∂x ∂y ∂z

is equivalent with the given differential equation; that is, μ being a proper function of x, y, z, we assume that there exist equations

∂φ = μX, ∂φ = μY, ∂φ = μZ;

∂x ∂y ∂z

these equations require

∂ (μY) ≈ ∂ (μZ), &c.,

∂z ∂y

and hence

X( ∂Z − ∂Y ) + Y( ∂X − ∂Z ) + Z( ∂Y − ∂X ) = 0;

∂y ∂z ∂z ∂x ∂x ∂y

conversely it can be proved that this is sufficient in order that μ may exist to render μ(Xdx + Ydy + Zdz) a perfect differential; in particular it may be satisfied in virtue of the three equations such as

∂Z − ∂Y = 0;

∂y ∂z

in which case we may take μ = 1. Assuming the condition in its general form, take in the given differential equation a plane section of the surface φ = C parallel to the plane z, viz. put z constant, and consider the resulting differential equation in the two variables x, y, namely Xdx + Ydy = 0; let ψ(x, y, z) = constant, be its integral, the constant z entering, as a rule, in ψ because it enters in X and Y. Now differentiate the relation ψ(x, y, z) = ƒ(z), where ƒ is a function to be determined, so obtaining

∂ψ dx + ∂ψ dy + ( ∂ψ − dƒ ) dz = 0;

∂x ∂y ∂z dz

there exists a function σ of x, y, z such that

∂ψ = σX, ∂ψ = σY,

∂x ∂y

because ψ = constant, is the integral of Xdx + Ydy = 0; we desire to prove that ƒ can be chosen so that also, in virtue of ψ(x, y, z) = ƒ(z), we have

∂ψ − dƒ = σZ, namely dƒ = ∂ψ − σZ;

∂z dz dz ∂z

if this can be proved the relation ψ(x, y, z) − ƒ(z) = constant, will be the integral of the given differential equation. To prove this it is enough to show that, in virtue of ψ(x, y, z) = ƒ(z), the function ∂ψ/∂x − σZ can be expressed in terms of z only. Now in consequence of the originally assumed relations,

∂ψ = μX, ∂φ = μY, ∂φ = μZ,

∂x ∂y ∂z

we have

∂ψ / ∂φ = σ = ∂ψ / ∂φ ,

∂x ∂x μ ∂y ∂y

and hence

∂ψ ∂φ − ∂ψ ∂φ = 0;

∂x ∂y ∂y ∂x

this shows that, as functions of x and y, ψ is a function of φ (see the note at the end of part i. of this article, on Jacobian determinants), so that we may write ψ = F(z, φ), from which

σ = ∂F ; then ∂ψ = ∂F + ∂F ∂φ = ∂F + σ · μZ = ∂F + σZ or ∂ψ − σZ = ∂F ;

μ ∂φ ∂z ∂z ∂φ ∂z ∂z μ ∂z ∂z ∂z

in virtue of ψ(x, y, z) = ƒ(z), and ψ = F(z, φ), the function φ can be written in terms of z only, thus ∂F/∂z can be written in terms of z only, and what we required to prove is proved.

Consider lastly a simple type of differential equation containing two independent variables, say x and y, and one dependent variable z, namely the equation

P ∂z + Q ∂z = R,

∂x ∂y

where P, Q, R are functions of x, y, z. This is known as Lagrange’s linear partial differential equation of the first order. To integrate this, consider first the ordinary differential equations dx/dz = P/R, dy/dz = Q/R, and suppose that two functions u, v, of x, y, z can be determined, independent of one another, such that the equations u = a, v = b, where a, b are arbitrary constants, lead to these ordinary differential equations, namely such that

P ∂u + Q ∂u + R ∂u = 0 and P ∂v + Q ∂v + R ∂v = 0.

∂x ∂y ∂z ∂x ∂y ∂z

Then if F(x, y, z) = 0 be a relation satisfying the original differential equations, this relation giving rise to

∂F + ∂F ∂z = 0 and ∂F + ∂F ∂z = 0, we have P ∂F + Q ∂F + R ∂F = 0.

∂x ∂z ∂x ∂y ∂z ∂y ∂x ∂y ∂z

It follows that the determinant of three rows and columns vanishes whose first row consists of the three quantities ∂F/∂x, ∂F/∂y, ∂F/∂z, whose second row consists of the three quantities ∂u/∂x, ∂u/∂y, ∂u/∂z, whose third row consists similarly of the partial derivatives of v. The vanishing of this so-called Jacobian determinant is known to imply that F is expressible as a function of u and v, unless these are themselves functionally related, which is contrary to hypothesis (see the note below on Jacobian determinants). Conversely, any relation φ(u, v) = 0 can easily be proved, in virtue of the equations satisfied by u and v, to lead to

P dz + Q dz = R.

dx dx

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

∂q = ∂Q ∂p and ∂q = ∂Q ∂p + ∂Q ;

∂x ∂p ∂x ∂y ∂p ∂y ∂y

these give

∂p ∂q − ∂p ∂q = ∂p ∂Q ;

∂x ∂y ∂y ∂x ∂x ∂y

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

∂u ∂v − ∂u ∂v ;

∂y ∂z ∂z ∂y

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

∂u ∂v − ∂u ∂v .

∂y ∂z ∂z ∂y

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

∂u ∂v − ∂u ∂v

∂y ∂z ∂z ∂y

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

∂u ∂v − ∂u ∂v

∂y ∂z ∂z ∂y

was zero), and so write F in the form F(x, y, z) = Φ(u, v, x). We then have

∂F = ∂Φ ∂u + ∂Φ ∂v + ∂Φ , ∂F = ∂Φ ∂u + ∂Φ ∂v , ∂F = ∂Φ ∂u + ∂Φ ∂v ;

∂x ∂u ∂x ∂v ∂x ∂x ∂y ∂u ∂y ∂v ∂y ∂z ∂u ∂z ∂v ∂z

thereby the Jacobian determinant of F, u, v is reduced to

∂Φ ( ∂u ∂v − ∂u ∂v );

∂x ∂y ∂z ∂z ∂y

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 (see Function: 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 ... ƒk of 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 = ck we can express x1, ... xk by convergent power series in the differences xk+1 − xºk+1, ... xn − xnº, and so regard x1, ... xk as functions of the remaining variables. This we often express by saying that the equations ƒ1 = c1, ... ƒk = ck can 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, ... φn of the (n + 1) variables x1, ... xnt be developable Ordinary 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, ... xºn when t = t0; and the only functions satisfying the equations and reducing respectively to xº1, ... xºn when t = t0, are those determined by continuation of these series. If the result of solving these n equations for xº1, ... xºn be 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, ... xn when t = t0. And in fact it has no other integrals so reducing. Thus this equation also possesses a unique integral reducing when t = t0 to an arbitrary function ψ(x1, ... xn), this integral being. ψ(ω1, ... ωn). Conversely the existence of these principal integrals ω1, ... ωn of 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 P0 are 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+1 an integral aggregate with real positive coefficients of the coefficients in Ps, Ps-1, ..., P0 and 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+1 in terms of the coefficients in ps, ps-1, ..., p0 and the coefficients in a and b. Hence as the coefficients in P0 and 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+1 is, 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 P0 are 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

F = r − (r − x − y) [ 1 − 4MR (1 − x + y ) −2 log ( 1 − t ) −1 ] 1/2 ,

r r R

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 that Simultaneous 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-called complete system is 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 the Complete 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, ... xn when in them for x1, ... xr are respectively put xº1, ... xºr; so that also the equations have in common a solution reducing when x1 = xº1, ... xr = xºr to 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, ... xn we 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 respectively Jacobian systems. to x2, ... xn when

x1 = xº1,

and its form shows that of these the first r − 1 are exactly x2 ... xr. Let these n − 1 functions together with x1 be 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 to System of total differential equations. the r(n − r) equations dxj/dxσ = cσr. That consistent with them we may be able to regard xr+1, ... xn as functions of x1, ... xr, these being regarded as independent variables, it is clearly necessary that when we differentiate cσj in regard to xρ on this hypothesis the result should be the same as when we differentiate cρj, in regard to xσ on this hypothesis. The differential coefficient of a function ƒ of x1, ... xn on this hypothesis, in regard to xρj is, 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, ... ωn be the principal solutions of the Jacobian system, Qσƒ = 0, reducing respectively to xr+1, ... xn when x1 = xº1, ... xr = xºr, and the equations ωr+1 = x0r+1, ... ωn = xºn be solved for xr+1, ... xn to give xj = ψj(x1, ... xr, x0r+1, ... xºn), these values solve the total equations and reduce respectively to x0r+1, ... xºn when 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 heading Pfaffian 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 particular Geometrical 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 ∞2 directions, 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 d1z0 as 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, δ1z0 ought to be equal to d1z0. But we find

a0dx0 + b0dy0 + dx0dy0( db + a0 db ),

dx0 dz0

and so at once reach the condition of integrability. If now we put x = 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 z0 for 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, ... xºn Mayer’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, ... mr as 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

dƒ/dt + Σ n (c1j + m2c2j + ... + mrcrj) dƒ/dxj = 0

j=r+1

in the n − r + 1 variables t, xr+1, ... xn. Determining the solutions Ωr+1, ... Ωn which reduce to respectively xr+1, ... xn when 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, ... mr in 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-called Pfaffian Expression adx + 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 + cu12 is 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, ... an be any functions of x1, ... xn, the so-called Pfaffian expression a1dx1 + ... + andxn can 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 + ... + umdtm can be put into the form u′1dt′1 + ... + u′mdt′m. (2) If the quantities u1, ..., u1, t1, ... tm be connected by a relation, the expression n1dt1 + ... + umdtm can be put into the format dt′ + u′1dt′1 + ... + u′m-1dt′m-1; and if the relation connecting u1, um, t1, ... tm be homogeneous in u1, ... um, then t′ can be taken to be zero. These two results are deductions from the theory of contact 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 made a 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 ..., tm are, except for this equation, independent variables. This is to be satisfied in virtue of Single linear Pfaffian equation. one or several relations connecting the variables; these must involve relations connecting t, t1, ... tm only, 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 ... tm only 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, ... tm only, 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, ... an are 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′m only.

This method for a single Pfaffian equation can, strictly speaking, be generalized to a simultaneous system of (n − r) Pfaffian equations dxj = c1jdx1 + ... + crjdxr only in the case already treated, Simultaneous Pfaffian equations. when this system is satisfied by regarding xr+1, ... xn as 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 2n Contact transformations. independent variables x1, ... xn, p1, ... pn we denote by (φψ) the sum of the n terms such as dφdψ/dpidxi − dψdφ/dpidxi For two functions of the (2n + 1) independent variables z, x1, ... xn, p1, ... pn we denote by φψ the sum of the n terms such as

dφ ( dψ + pi dψ ) − dψ ( dφ pi dφ ).

dpi dxi dz dpi dxi dz

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, ... Pn be such functions Of x1, ... xn, p1 ... pn that P1dX1 + ... + PndXn is identically equal to p1dx1 + ... + pndxn, it can be shown by elementary algebra, after equating coefficients of independent differentials, (1) that the functions X1, ... Pn are independent functions of the 2n variables x1, ... pn, so that the equations x′i = Xi, p′i = Pi can be solved for x1, ... xn, p1, ... pn, and represent therefore a transformation, which we call a homogeneous contact transformation; (2) that the X1, ... Xn are homogeneous functions of p1, ... pn of zero dimensions, the P1, ... Pn are homogeneous functions of p1, ... pn of 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, ... Xn be independent functions, each homogeneous of zero dimension in p1, ... pn satisfying the ½n(n − 1) relations (XiXj) = 0, then P1, ... Pn can 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, Qi for -Pi/Pn+1, for i = 1, ... n, put qi for -pi/pn+1 and σ for qn+1/Qn+1, and then finally write P1, ... Pn, p1, ... pn for Q1, ... Qn, q1, ... qn, we obtain the following results: If ZX1 ... Xn, P1, ... Pn be functions of z, x1, ... xn, p1, ... pn, such that the expression dZ − P1dX1 − ... − PndXn is identically equal to σ(dz − p1dx1 − ... − pndxn), and σ not zero, then (1) the functions Z, X1, ... Xn, P1, ... Pn are independent functions of z, x1, ... xn, p1, ... pn, so that the equations z′ = Z, x′i = Xi, p′i = Pi can be solved for z, x1, ... xn, p1, ... pn and determine a transformation which we call a (non-homogeneous) contact transformation; (2) the Z, X1, ... Xn verify the ½n(n + 1) identities [ZXi] = 0, [XiXj] = 0. And the further identities

[PiXi] = σ, [PiXj] = 0, [PiZ] = σPi, [PiPj] = 0,

[Zσ] = σ dZ − σ², [Xiσ] = σ dXi , [Piσ] = dPi

dz dz dz

are also verified. Conversely, if Z, x1, ... Xn be independent functions satisfying the identities [ZXi] = 0, [XiXj] = 0, then σ, other than zero, and P1, ... Pn can 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, ... Pn be 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, ... Xn are 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, ... Xn be 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, ... Pn can be found, by solution of linear algebraic equations, to render true the identity dU + P1dX1 + ... + PndXn = p1dx1 + ... + pndxn; (3) Functions X1, ... Xn, P1, ... Pn can 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, ... Xr 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 Partial 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 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, ... 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 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 Order 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 under Pfaffian 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

dx′/ dF′ = dy′/ dF′ = dz′/ ( p′ dF′ + q′ dF′ ).

dp′ dq′ dp′ dq′

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

dx′/ dF′ = dy′/ dF′ = dz′/ ( p′ dF′ + q′ dF′ ) = dp′/ ( − dF′ − p′ dF′ ) = dq′/ ( − dF′ − q′ dF′ )

dp′ dq′ dp′ dq′ dx′ dz′ dy′ dz′

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

δx′( dF′ + p′ dF′ ) + δy′( dF′ q′ dF′ ) = 0;

dx′ dz′ dy′ dz′

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 a characteristic 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 ∞4 elements satisfying F = 0.