It is easy to see that the series for ℜ(u), u−2+ Σ′[(u + mω + m′ω′)2− (mω + m′ω′)2], is unaffected by replacing ω, ω′ by two quantities Ω, Ω′ equal respectively to pω + qω′, p′ω′ + q′ω′, where p, q, p′, q′ are any integers for which pq′ − p′q = ±1; further it can be proved that all substitutions with integer coefficients Ω = pω + qω′, Ω′ = p′ω + q′ω′, wherein pq′ − p′q = 1, can be built up by repetitions of the two particular substitutions (Ω = −ω′, Ω′ = ω), (Ω = ω, Ω′ = ω + ω′). Consider the function of the ratio ω′/ω expressed byh = −ℜ (½ω′) / ℜ(½ω);it is at once seen from the properties of the function ℜ(u) that by the two particular substitutions referred to we obtain the corresponding substitutions for h expressed byh′ = 1/h, h′ = 1 − h;thus, by all the integer substitutions Ω = pω + qω′, Ω′ = p′ω + q′ω′, in which pq′ − p′q = 1, the function h can only take one of the six values h, 1/h, 1 − h, 1/(1 − h), h/(h − 1), (h − 1)/h, which are the roots of an equation in θ,(1 − θ + θ2)3=(1 − h + h2)3;θ2(1 − θ)2h2(1 − h)2the function of τ, = ω′/ω, expressed by the right side, is thus unaltered by every one of the substitutions τ′ = (p′ + q′τ / p + qτ), wherein p, q, p′, q′ are integers having pq′ − p′q = 1. If the imaginary part σ, of τ, which we may write τ = ρ + iσ, is positive, the imaginary part of τ′, which is equal to σ(pq′ − p′q)/[(p + qρ)2+ q2σ2], is also positive; suppose σ to be positive; it can be shown that the upper half of the infinite plane of the complex variable τ can be divided into regions, all bounded by arcs of circles (or straight lines), no two of these regions overlapping, such that any substitution of the kind under consideration, τ′ = (p′ + q′τ)/(p + qτ) leads from an arbitrary point τ, of one of these regions, to a point τ′ of another; taking τ = ρ + iσ, one of these regions may be taken to be that for which −½ < ρ < ½, ρ2+ σ2> 1, together with the points for which ρ is negative on the curves limiting this region; then every other region is obtained from this so-called fundamental region by one and only one of the substitutions τ = (p′ + q′τ)/(p + qτ), and hence by a definite combination of the substitutions τ′ = −1/τ, τ′ = 1 + τ. Upon the infinite half plane of τ, the function considered above,z(τ) =4⁄27[ℜ2(½ω) + ℜ (z(½ω) ℜ (½ω′) + ℜ2(½ω′)]3ℜ2(½ω) ℜ2(½ω′) [ℜ (½ω) + ℜ (½ω′]2is a single valued monogenic function, whose only essential singularities are the points τ′ = (p′ + q′τ)/(p + qτ) for which τ = ∞, namely those for which τ′ is any real rational value; the real axis is thus a line over which the function z(τ) cannot be continued, having an essential singularity in every arc of it, however short; in the fundamental region, z(τ) has thus only the single essential singularity, r = ρ + iσ, where σ = ∞; in this fundamental region z(τ) takes any assigned complex value just once, the relation z(τ′) = z(τ) requiring, as can be shown, that τ′ is of the form (p′ + q′τ)/(p + qτ), in which p, q, p′, q′ are integers with pq′ − p′q = 1; the function z(τ) has thus a similar behaviour in every other of the regions. The division of the plane into regions is analogous to the division of the plane, in the case of doubly periodic functions, into parallelograms; in that case we considered only functions without essential singularities, and in each of the regions the function assumed every complex value twice, at least. Putting, as another function of τ, J(τ) = z(τ) [z(τ) − 1], it can be shown that J(τ) = 0 for τ = exp (2⁄3πi), that J(τ) = 1 for τ = i, these being values of τ on the boundary of the fundamental region; like z(τ) it has an essential singularity for τ = ρ + iσ, σ = + ∞. In thetheory of linear differential equations it is important to consider the inverse function τ(J); this is infinitely many valued, having a cycle of three values for circulation of J about J = 0 (the circuit of this point leading to a linear substitution for τ of period 3, such as τ′ = −(1 + τ)−1), having a cycle of two values about J = 1 (the circuit leading to a linear substitution for τ of period 2, such as τ′ = −τ−1), and having a cycle of infinitely many values about J = ∞ (the circuit leading to a linear substitution for τ which is not periodic, such as τ′ = 1 + τ). These are the only singularities for the function τ(J). Each of the functions[J(τ)]1/3, [J(τ) − 1]1/2,[−ℜ (½ω) + 2ℜ (½ω′)]1/8,ℜ (½ω) − ℜ (½)ω′)beside many others (see below), is a single valued function of τ, and is expressible without ambiguity in terms of the single valued function of τ,η(τ) = exp(iπτ) Π∞n=1[1 − exp (2iπnτ)] = exp(iπτ) Σ∞m=−∞(−1)mexp [(3m2+ m) iπτ].1212It should be remarked, however, that η(τ) is not unaltered by all the substitutions we have considered; in factη(−τ−1) = (−iτ) ½η (τ), η(1 + τ) = exp (1⁄12iπ) η(τ).The aggregate of the substitutions τ′ = (p′ + q′τ)/(p + qτ), wherein p, q, p′, q′ are integers with pq′ − p′q = 1, represents aGroup; the function J(τ), unaltered by all these substitutions, is called aModular Function. More generally any function unaltered by all the substitutions of a group of linear substitutions of its variable is called anAutomorphic Function. A rational function, of its variable h, of this character, is the function (1 − h + h2)3h−2(1 − h)−2presenting itself incidentally above; and there are other rational functions with a similar property, the group of substitutions belonging to any one of these being, what is a very curious fact, associable with that of the rotations of one of the regular solids, about an axis through its centre, which bring the solid into coincidence with itself. Other automorphic functions are the double periodic functions already discussed; these, as we have seen, enable us to solve the algebraic equation y2= 4x3− g2x − g3(and in fact many other algebraic equations, see below, under § 23,Geometrical Applications of Elliptic Functions) in terms of single valued functions x = ℜ(u), y = −ℜ′(u). A similar utility, of a more extended kind, belongs to automorphic functions in general; but it can be shown that such functions necessarily have an infinite number of essential singularities except for the simplest cases.The modular function J(τ) considered above, unaltered by the group of linear substitutions τ′ = (p′ + q′τ) / (p + qτ), where p, q, p′, q′ are integers with pq′ − p′q = 1, may be taken as the independent variable x of a differential equation of the third order, of the forms″′−3(s″)2=1 − α2+1 − β2+α2+ β2− γ2− 1,s′2s′2(x − 1)22x22x (x − 1)where s′ = ds/dx, &c., of which the dependent variable s is equal to τ. A differential equation of this form is satisfied by the quotient of two independent integrals of the linear differential equation of the second order satisfied by the hypergeometric functions. If the solution of the differential equation for s be written s(α,β,γ, x), we have in fact τ = s(½,1⁄3, 0, J). If we introduce also the function of τ given byλ =2ℜ (½ω′) + ℜ (½ω),ℜ (½ω′) − ℜ (½ω)we similarly have τ = s(0, 0, 0, λ); this function λ is a single valued function of τ, which is also a modular function, being unaltered by a group of integral substitutions also of the form τ′ = (p′ + q′τ)/(p + qτ), with pq′ − p′q = 1, but with the restriction that p′ and q are even integers, and therefore p and q′ are odd integers. This group is thus a subgroup of the general modular group, and is in fact of the kind called a self-conjugate subgroup. As in the general case this subgroup is associated with a subdivision of the plane into regions of which any one is obtained from a particular region, called the fundamental region, by a particular one of the substitutions of the subgroup. This fundamental region, putting τ = ρ + iσ, may be taken to be that given by −1 < ρ < 1, (ρ + ½)2+ σ2> ¼, (ρ − ½)2+ σ2> ¼, and is built up of six of the regions which arose for the general modular group associated with J(τ). Within this fundamental region, λ takes every complex value just once, except the values λ = 0, 1, ∞, which arise only at the angular points τ = 0, τ = ∞, τ = − 1 and the equivalent point τ = 1; these angular points are essential singularities for the function λ(τ). For λ(τ) as for J(τ), the region of existence is the upper half plane of τ, there being an essential singularity in every length of the real axis, however short.If, beside the plane of τ, we take a plane to represent the values of λ, the function τ = s(0, 0, 0, λ) being considered thereon, the values of τ belonging to the interior of the fundamental region of the τ-plane considered above, will require the consideration of the whole of the λ-plane taken once with the exception of the portions of the real axis lying between −∞ and 0 and between 1 and +∞, the two sides of the first portion corresponding to the circumferences of the τ-plane expressed by (ρ + ½)2+ σ2= ¼, (ρ − ½)2+ σ2= ¼, while the two sides of the latter portion, for which λ is real and > 1, correspond to the lines of the τ-plane expressed by ρ = ±1. The line for which λ is real, positive and less than unity corresponds to the imaginary axis of the τ-plane, lying in the interior of the fundamental region. All the values of τ = s(0, 0, 0, λ) may then be derived from those belonging to the fundamental region of the τ-plane by making λ describe a proper succession of circuits about the points λ = 0, λ = 1; any such circuit subjects τ to a linear substitution of the subgroup of τ considered, and corresponds to a change of τ from a point of the fundamental region to a corresponding point of one of the other regions.
It is easy to see that the series for ℜ(u), u−2+ Σ′[(u + mω + m′ω′)2− (mω + m′ω′)2], is unaffected by replacing ω, ω′ by two quantities Ω, Ω′ equal respectively to pω + qω′, p′ω′ + q′ω′, where p, q, p′, q′ are any integers for which pq′ − p′q = ±1; further it can be proved that all substitutions with integer coefficients Ω = pω + qω′, Ω′ = p′ω + q′ω′, wherein pq′ − p′q = 1, can be built up by repetitions of the two particular substitutions (Ω = −ω′, Ω′ = ω), (Ω = ω, Ω′ = ω + ω′). Consider the function of the ratio ω′/ω expressed by
h = −ℜ (½ω′) / ℜ(½ω);
it is at once seen from the properties of the function ℜ(u) that by the two particular substitutions referred to we obtain the corresponding substitutions for h expressed by
h′ = 1/h, h′ = 1 − h;
thus, by all the integer substitutions Ω = pω + qω′, Ω′ = p′ω + q′ω′, in which pq′ − p′q = 1, the function h can only take one of the six values h, 1/h, 1 − h, 1/(1 − h), h/(h − 1), (h − 1)/h, which are the roots of an equation in θ,
the function of τ, = ω′/ω, expressed by the right side, is thus unaltered by every one of the substitutions τ′ = (p′ + q′τ / p + qτ), wherein p, q, p′, q′ are integers having pq′ − p′q = 1. If the imaginary part σ, of τ, which we may write τ = ρ + iσ, is positive, the imaginary part of τ′, which is equal to σ(pq′ − p′q)/[(p + qρ)2+ q2σ2], is also positive; suppose σ to be positive; it can be shown that the upper half of the infinite plane of the complex variable τ can be divided into regions, all bounded by arcs of circles (or straight lines), no two of these regions overlapping, such that any substitution of the kind under consideration, τ′ = (p′ + q′τ)/(p + qτ) leads from an arbitrary point τ, of one of these regions, to a point τ′ of another; taking τ = ρ + iσ, one of these regions may be taken to be that for which −½ < ρ < ½, ρ2+ σ2> 1, together with the points for which ρ is negative on the curves limiting this region; then every other region is obtained from this so-called fundamental region by one and only one of the substitutions τ = (p′ + q′τ)/(p + qτ), and hence by a definite combination of the substitutions τ′ = −1/τ, τ′ = 1 + τ. Upon the infinite half plane of τ, the function considered above,
is a single valued monogenic function, whose only essential singularities are the points τ′ = (p′ + q′τ)/(p + qτ) for which τ = ∞, namely those for which τ′ is any real rational value; the real axis is thus a line over which the function z(τ) cannot be continued, having an essential singularity in every arc of it, however short; in the fundamental region, z(τ) has thus only the single essential singularity, r = ρ + iσ, where σ = ∞; in this fundamental region z(τ) takes any assigned complex value just once, the relation z(τ′) = z(τ) requiring, as can be shown, that τ′ is of the form (p′ + q′τ)/(p + qτ), in which p, q, p′, q′ are integers with pq′ − p′q = 1; the function z(τ) has thus a similar behaviour in every other of the regions. The division of the plane into regions is analogous to the division of the plane, in the case of doubly periodic functions, into parallelograms; in that case we considered only functions without essential singularities, and in each of the regions the function assumed every complex value twice, at least. Putting, as another function of τ, J(τ) = z(τ) [z(τ) − 1], it can be shown that J(τ) = 0 for τ = exp (2⁄3πi), that J(τ) = 1 for τ = i, these being values of τ on the boundary of the fundamental region; like z(τ) it has an essential singularity for τ = ρ + iσ, σ = + ∞. In thetheory of linear differential equations it is important to consider the inverse function τ(J); this is infinitely many valued, having a cycle of three values for circulation of J about J = 0 (the circuit of this point leading to a linear substitution for τ of period 3, such as τ′ = −(1 + τ)−1), having a cycle of two values about J = 1 (the circuit leading to a linear substitution for τ of period 2, such as τ′ = −τ−1), and having a cycle of infinitely many values about J = ∞ (the circuit leading to a linear substitution for τ which is not periodic, such as τ′ = 1 + τ). These are the only singularities for the function τ(J). Each of the functions
beside many others (see below), is a single valued function of τ, and is expressible without ambiguity in terms of the single valued function of τ,
It should be remarked, however, that η(τ) is not unaltered by all the substitutions we have considered; in fact
η(−τ−1) = (−iτ) ½η (τ), η(1 + τ) = exp (1⁄12iπ) η(τ).
The aggregate of the substitutions τ′ = (p′ + q′τ)/(p + qτ), wherein p, q, p′, q′ are integers with pq′ − p′q = 1, represents aGroup; the function J(τ), unaltered by all these substitutions, is called aModular Function. More generally any function unaltered by all the substitutions of a group of linear substitutions of its variable is called anAutomorphic Function. A rational function, of its variable h, of this character, is the function (1 − h + h2)3h−2(1 − h)−2presenting itself incidentally above; and there are other rational functions with a similar property, the group of substitutions belonging to any one of these being, what is a very curious fact, associable with that of the rotations of one of the regular solids, about an axis through its centre, which bring the solid into coincidence with itself. Other automorphic functions are the double periodic functions already discussed; these, as we have seen, enable us to solve the algebraic equation y2= 4x3− g2x − g3(and in fact many other algebraic equations, see below, under § 23,Geometrical Applications of Elliptic Functions) in terms of single valued functions x = ℜ(u), y = −ℜ′(u). A similar utility, of a more extended kind, belongs to automorphic functions in general; but it can be shown that such functions necessarily have an infinite number of essential singularities except for the simplest cases.
The modular function J(τ) considered above, unaltered by the group of linear substitutions τ′ = (p′ + q′τ) / (p + qτ), where p, q, p′, q′ are integers with pq′ − p′q = 1, may be taken as the independent variable x of a differential equation of the third order, of the form
where s′ = ds/dx, &c., of which the dependent variable s is equal to τ. A differential equation of this form is satisfied by the quotient of two independent integrals of the linear differential equation of the second order satisfied by the hypergeometric functions. If the solution of the differential equation for s be written s(α,β,γ, x), we have in fact τ = s(½,1⁄3, 0, J). If we introduce also the function of τ given by
we similarly have τ = s(0, 0, 0, λ); this function λ is a single valued function of τ, which is also a modular function, being unaltered by a group of integral substitutions also of the form τ′ = (p′ + q′τ)/(p + qτ), with pq′ − p′q = 1, but with the restriction that p′ and q are even integers, and therefore p and q′ are odd integers. This group is thus a subgroup of the general modular group, and is in fact of the kind called a self-conjugate subgroup. As in the general case this subgroup is associated with a subdivision of the plane into regions of which any one is obtained from a particular region, called the fundamental region, by a particular one of the substitutions of the subgroup. This fundamental region, putting τ = ρ + iσ, may be taken to be that given by −1 < ρ < 1, (ρ + ½)2+ σ2> ¼, (ρ − ½)2+ σ2> ¼, and is built up of six of the regions which arose for the general modular group associated with J(τ). Within this fundamental region, λ takes every complex value just once, except the values λ = 0, 1, ∞, which arise only at the angular points τ = 0, τ = ∞, τ = − 1 and the equivalent point τ = 1; these angular points are essential singularities for the function λ(τ). For λ(τ) as for J(τ), the region of existence is the upper half plane of τ, there being an essential singularity in every length of the real axis, however short.
If, beside the plane of τ, we take a plane to represent the values of λ, the function τ = s(0, 0, 0, λ) being considered thereon, the values of τ belonging to the interior of the fundamental region of the τ-plane considered above, will require the consideration of the whole of the λ-plane taken once with the exception of the portions of the real axis lying between −∞ and 0 and between 1 and +∞, the two sides of the first portion corresponding to the circumferences of the τ-plane expressed by (ρ + ½)2+ σ2= ¼, (ρ − ½)2+ σ2= ¼, while the two sides of the latter portion, for which λ is real and > 1, correspond to the lines of the τ-plane expressed by ρ = ±1. The line for which λ is real, positive and less than unity corresponds to the imaginary axis of the τ-plane, lying in the interior of the fundamental region. All the values of τ = s(0, 0, 0, λ) may then be derived from those belonging to the fundamental region of the τ-plane by making λ describe a proper succession of circuits about the points λ = 0, λ = 1; any such circuit subjects τ to a linear substitution of the subgroup of τ considered, and corresponds to a change of τ from a point of the fundamental region to a corresponding point of one of the other regions.
§ 22.A Property of Integral Functions deduced from the Theory of Modular Functions.—Consider now the function exp(z), for finite values of z; for such values of z, exp(z) never vanishes, and it is impossible to assign a closed circuit for z in the finite part of the plane of z which will make the function λ = exp(z) pass through a closed succession of values in the plane of λ having λ = 0 in its interior; the function s[0, 0, 0, exp(z)], however z vary in the finite part of the plane, will therefore never be subjected to those linear substitutions imposed upon s(0, 0, 0, λ) by a circuit of λ about λ = 0; more generally, if φ(z) be an integral function of z, never becoming either zero or unity for finite values of z, the function λ = φ(z), however z vary in the finite part of the plane, will never make, in the plane of λ, a circuit about either λ = 0 or λ = 1, and s(0, 0, 0, λ), that is s[0, 0, 0, φ(z)], will be single valued for all finite values of z; it will moreover remain finite, and be monogenic. In other words, s[0, 0, 0, φ(z)] is also an integral function—whose imaginary part, moreover, by the property of s(0, 0, 0, λ), remains positive for all finite values of z. In that case, however, exp {is[0, 0, 0, φ(z)]} would also be an integral function of z with modulus less than unity for all finite values of z. If, however, we describe a circle of radius R in the z plane, and consider the greatest value of the modulus of an integral function upon this circle, this certainly increases indefinitely as R increases. We can infer therefore thatan integral function φ(z) which does not vanish for any finite value of z, takes the value unity and hence(by considering the function A−1φ(z))takes every other value for some definite value of z; or, an integral function for which both the equations φ(z) = A, φ(z) = B are unsatisfied by definite values of z, does not exist, A and B being arbitrary constants.
A similar theorem can be proved in regard to the values assumed by the function φ(z) for points z of modulus greater than R, however great R may be, also with the help of modular functions. In general terms it may be stated that it is a very exceptional thing for an integral function not to assume every complex value an infinite number of times.Another application of modular functions is to prove that the function s(α, β, γ, λ) is a single valued function of τ = s(0, 0, 0, λ); for, putting τ′ = (τ − i)/(τ + i), the values of τ′ which correspond to the singular points λ = 0, 1, ∞ of s(α, β, γ, λ), though infinite in number, all lie on the circumference of the circle |τ′| = 1, within which therefore s(α, β, γ, x) is expressible in a formΣ∞n=0anτ′n. More generally any monogenic function of λ which is single valued save for circuits of the points λ = 0, 1, ∞, is a single valued function of τ = s(0, 0, 0, λ). Identifying λ with the square of the modulus in Legendre’s form of the elliptical integral, we have τ = iK′/K, whereK =∫10dt, K′ =∫10dt;√[1 − t2] [1 − λt2]√[1 − t2] [1 − (1 − λ) t2]functions such as λ1/4, (1 − λ)1/4, [λ(1 − λ)]1/4, which have only λ = 0, 1, ∞ as singular points, were expressed by Jacobi as power series in q = eiπτ, and therefore, at least for a limited range of values of τ, as single valued functions of τ; it follows by the theorem given that any product of a root of λ and a root of 1 − λ is a single valued function of τ. More generally the differential equationx(1 − x)d2y+ [γ − (α + β + 1)x]dy− αβγ = 0dx2dxmay be solved by expressing both the independent and dependent variables as single valued functions of a single variable τ, the expression for the independent variable being x = λ(τ).
A similar theorem can be proved in regard to the values assumed by the function φ(z) for points z of modulus greater than R, however great R may be, also with the help of modular functions. In general terms it may be stated that it is a very exceptional thing for an integral function not to assume every complex value an infinite number of times.
Another application of modular functions is to prove that the function s(α, β, γ, λ) is a single valued function of τ = s(0, 0, 0, λ); for, putting τ′ = (τ − i)/(τ + i), the values of τ′ which correspond to the singular points λ = 0, 1, ∞ of s(α, β, γ, λ), though infinite in number, all lie on the circumference of the circle |τ′| = 1, within which therefore s(α, β, γ, x) is expressible in a formΣ∞n=0anτ′n. More generally any monogenic function of λ which is single valued save for circuits of the points λ = 0, 1, ∞, is a single valued function of τ = s(0, 0, 0, λ). Identifying λ with the square of the modulus in Legendre’s form of the elliptical integral, we have τ = iK′/K, where
functions such as λ1/4, (1 − λ)1/4, [λ(1 − λ)]1/4, which have only λ = 0, 1, ∞ as singular points, were expressed by Jacobi as power series in q = eiπτ, and therefore, at least for a limited range of values of τ, as single valued functions of τ; it follows by the theorem given that any product of a root of λ and a root of 1 − λ is a single valued function of τ. More generally the differential equation
may be solved by expressing both the independent and dependent variables as single valued functions of a single variable τ, the expression for the independent variable being x = λ(τ).
§ 23.Geometrical Applications of Elliptic Functions.—Consider any irreducible algebraic equation rational in x, y, f(x, y) = 0, of such a form that the equation represents a plane curve of order n with ½n(n − 3) double points; taking upon this curve n− 3 arbitrary fixed points, draw through these and the double points the most general curve of order n − 2; this will intersectƒ in n(n − 2) − n(n − 3) − (n − 3) = 3 other points, and will contain homogeneously at least ½(n − 1)n − ½n(n − 3) −(n − 3) = 3 arbitrary constants, and so will be of the form λφ + λ1φ1+ λ2φ2+ ... = 0, wherein λ3, λ4, ... are in general zero. Put now ξ = φ1/φ, η = φ2/φ and eliminate x, y between these equations and ƒ(x, y) = 0, so obtaining a rational irreducible equation F(ξ, η) = 0, representing a further plane curve. To any point (x, y) of ƒ will then correspond a definite point (ξ, η) of F.
For a general position of (x, y) upon ƒ the equations φ1(x′, x′)/φ(x′, x′) = φ1(x, y)/φ(x, y), φ2(x′, x′)/φ(x′, x′) = φ2(x, y)/φ(x, y), subject to ƒ(x′, x′) = 0, will have the same number of solutions (x′, x′); if their only solution is x′ = x, x′ = y, then to any position (ξ, η) of F will conversely correspond only one position (x, y) of ƒ. If these equations have another solution beside (x, y), then any curve λφ + λ1φ1+ λ2φ2= 0 which passes (through the double points of ƒ and) through the n − 2 points of ƒ constituted by the fixed n− 3 points and a point (x0, y0), will necessarily pass through a further point, say (x0′, y0′), and will have only one further intersection with ƒ; such a curve, with the n − 2 assigned points, beside the double points, of ƒ, will be of the form μψ + μ1ψ1+ ... = 0, where μ2, μ3, ... are generally zero; considering the curves ψ + tψ1= 0, for variable t, one of these passes through a further arbitrary point of ƒ, by choosing t properly, and conversely an arbitrary value of t determines a single further point of ƒ; the co-ordinates of the points of ƒ are thus rational functions of a parameter t, which is itself expressible rationally by the co-ordinates of the point; it can be shown algebraically that such a curve has not ½(n − 3)n but ½(n − 3)n + 1 double points. We may therefore assume that to every point of F corresponds only one point of ƒ, and there is a birational transformation between these curves; the coefficients in this transformation will involve rationally the co-ordinates of the n− 3 fixed points taken upon ƒ, that is, at the least, by taking these to be consecutive points, will involve the co-ordinates of one point of ƒ, and will not be rational in the coefficients of ƒ unless we can specify a point of ƒ whose co-ordinates are rational in these. The curve F is intersected by a straight line aξ + bη + c = 0 in as many points as the number of unspecified intersections of ƒ with aφ + bφ1+ cφ2= 0, that is, 3; or F will be a cubic curve, without double points.Such a cubic curve has at least one point of inflection Y, and if a variable line YPQ be drawn through Y to cut the curve again in P and Q, the locus of a point R such that YR is the harmonic mean of YP and YQ, is easily proved to be a straight line. Take now a triangle of reference for homogeneous co-ordinates XYZ, of which this straight line is Y = 0, and the inflexional tangent at Y is Z = 0; the equation of the cubic curve will then be of the formZY² = aX³ + bX²Z + cXZ² + dZ³;by putting X equal to λX + μZ, that is, choosing a suitable line through Y to be X = 0, and choosing λ properly, this is reduced to the formZY² = 4X³ − g2XZ² − g3Z³,of which a representation is given, valid for every point, in terms of the elliptic functions ℜ(u), ℜ′(u), by taking X = Zℜ(u), Y = Zℜ′(u). The value of u belonging to any point is definite save for sums of integral multiples of the periods of the elliptic functions, being given byu =∫(x)(∞)ZdX − XdZ,ZYwhere (∞) denotes the point of inflection.It thus appears that the co-ordinates of any point of a plane curve, ƒ, of order n with ½(n − 3)n double points are expressible as elliptic functions, there being, save for periods, a definite value of the argument u belonging to every point of the curve. It can then be shown that if a variable curve, φ, of order m be drawn, passing through the double points of the curve, the values of the argument u at the remaining intersections of φ with ƒ, have a sum which is unaffected by variation of the coefficients of φ, save for additive aggregates of the periods. In virtue of the birational transformation this theorem can be deduced from the theorem that if any straight line cut the cubic y² = 4x³ − g2x − g3, in points (u1), (u2), (u3), the sum u1+ u2+ u3is zero, or a period; or the general theorem is a corollary from Abel’s theorem proved under § 17,Integrals of Algebraic Functions. To prove the result directly for the cubic we remark that the variation of one of the intersections (x, y) of the cubic with the straight line y = mx + n, due to a variation δm, δn in m and n, is obtained by differentiation of the equation for the three abscissae, namely the equationF(x) = 4x³ − g2x − g3− (mx + n)² = 0,and is thus given bydx=xδm + δn,yF′(x)and the sum of three such fractions as that on the right for the three roots of F(x) = 0 is zero; hence u1+ u2+ u3is independent of the straight line considered; if in particular this become the inflexional tangent each of u1, u2, u3vanishes. It may be remarked in passing that x1+ x2+ x3= ¼m², and hence is ¼ {(y1− y2)/(x1− x2)}²; so that we have another proof of the addition equation for the function ℜ(u). From this theorem for the cubic curve many of its geometrical properties, as for example those of its inflections, the properties of inscribed polygons, of the three kinds of corresponding points, and the theory of residuation, are at once obvious. And similar results hold for the curve of order n with ½(n − 3)n double points.
For a general position of (x, y) upon ƒ the equations φ1(x′, x′)/φ(x′, x′) = φ1(x, y)/φ(x, y), φ2(x′, x′)/φ(x′, x′) = φ2(x, y)/φ(x, y), subject to ƒ(x′, x′) = 0, will have the same number of solutions (x′, x′); if their only solution is x′ = x, x′ = y, then to any position (ξ, η) of F will conversely correspond only one position (x, y) of ƒ. If these equations have another solution beside (x, y), then any curve λφ + λ1φ1+ λ2φ2= 0 which passes (through the double points of ƒ and) through the n − 2 points of ƒ constituted by the fixed n− 3 points and a point (x0, y0), will necessarily pass through a further point, say (x0′, y0′), and will have only one further intersection with ƒ; such a curve, with the n − 2 assigned points, beside the double points, of ƒ, will be of the form μψ + μ1ψ1+ ... = 0, where μ2, μ3, ... are generally zero; considering the curves ψ + tψ1= 0, for variable t, one of these passes through a further arbitrary point of ƒ, by choosing t properly, and conversely an arbitrary value of t determines a single further point of ƒ; the co-ordinates of the points of ƒ are thus rational functions of a parameter t, which is itself expressible rationally by the co-ordinates of the point; it can be shown algebraically that such a curve has not ½(n − 3)n but ½(n − 3)n + 1 double points. We may therefore assume that to every point of F corresponds only one point of ƒ, and there is a birational transformation between these curves; the coefficients in this transformation will involve rationally the co-ordinates of the n− 3 fixed points taken upon ƒ, that is, at the least, by taking these to be consecutive points, will involve the co-ordinates of one point of ƒ, and will not be rational in the coefficients of ƒ unless we can specify a point of ƒ whose co-ordinates are rational in these. The curve F is intersected by a straight line aξ + bη + c = 0 in as many points as the number of unspecified intersections of ƒ with aφ + bφ1+ cφ2= 0, that is, 3; or F will be a cubic curve, without double points.
Such a cubic curve has at least one point of inflection Y, and if a variable line YPQ be drawn through Y to cut the curve again in P and Q, the locus of a point R such that YR is the harmonic mean of YP and YQ, is easily proved to be a straight line. Take now a triangle of reference for homogeneous co-ordinates XYZ, of which this straight line is Y = 0, and the inflexional tangent at Y is Z = 0; the equation of the cubic curve will then be of the form
ZY² = aX³ + bX²Z + cXZ² + dZ³;
by putting X equal to λX + μZ, that is, choosing a suitable line through Y to be X = 0, and choosing λ properly, this is reduced to the form
ZY² = 4X³ − g2XZ² − g3Z³,
of which a representation is given, valid for every point, in terms of the elliptic functions ℜ(u), ℜ′(u), by taking X = Zℜ(u), Y = Zℜ′(u). The value of u belonging to any point is definite save for sums of integral multiples of the periods of the elliptic functions, being given by
where (∞) denotes the point of inflection.
It thus appears that the co-ordinates of any point of a plane curve, ƒ, of order n with ½(n − 3)n double points are expressible as elliptic functions, there being, save for periods, a definite value of the argument u belonging to every point of the curve. It can then be shown that if a variable curve, φ, of order m be drawn, passing through the double points of the curve, the values of the argument u at the remaining intersections of φ with ƒ, have a sum which is unaffected by variation of the coefficients of φ, save for additive aggregates of the periods. In virtue of the birational transformation this theorem can be deduced from the theorem that if any straight line cut the cubic y² = 4x³ − g2x − g3, in points (u1), (u2), (u3), the sum u1+ u2+ u3is zero, or a period; or the general theorem is a corollary from Abel’s theorem proved under § 17,Integrals of Algebraic Functions. To prove the result directly for the cubic we remark that the variation of one of the intersections (x, y) of the cubic with the straight line y = mx + n, due to a variation δm, δn in m and n, is obtained by differentiation of the equation for the three abscissae, namely the equation
F(x) = 4x³ − g2x − g3− (mx + n)² = 0,
and is thus given by
and the sum of three such fractions as that on the right for the three roots of F(x) = 0 is zero; hence u1+ u2+ u3is independent of the straight line considered; if in particular this become the inflexional tangent each of u1, u2, u3vanishes. It may be remarked in passing that x1+ x2+ x3= ¼m², and hence is ¼ {(y1− y2)/(x1− x2)}²; so that we have another proof of the addition equation for the function ℜ(u). From this theorem for the cubic curve many of its geometrical properties, as for example those of its inflections, the properties of inscribed polygons, of the three kinds of corresponding points, and the theory of residuation, are at once obvious. And similar results hold for the curve of order n with ½(n − 3)n double points.
§ 24.Integrals of Algebraic Functions in Connexion with the Theory of Plane Curves.—The developments which have been explained in connexion with elliptic functions may enable the reader to appreciate the vastly more extensive theory similarly arising for any algebraical irrationality, ƒ(x, y) = o.
The algebraical integrals ∫ R(x, y)dx associated with this may as before be divided into those of thefirst kind, which have no infinities, those of thesecond kind, possessing only algebraical infinities, and those of thethird kind, for which logarithmic infinities enter. Here there is a certain number, p, greater than unity, of linearly independent integrals of the first kind; and this number p is unaltered by any birational transformation of the fundamental equation ƒ(x, y) = 0; a rational function can be constructed with poles of the first order at p + 1 arbitrary positions (x, y), satisfying ƒ(x, y) = 0, but not with a fewer number unless their positions are chosen properly, a property we found for the case p = 1; and p is the number of linearly independent curves of order n − 3 passing through the double points of the curve of order n expressed by ƒ(x, y) = 0. Again any integral of the second kind can be expressed as a sum of p integrals of this kind, with poles of the first order at arbitrary positions, together with rational functions and integrals of the first kind; and an integral of the second kind can be found with one pole of the first order of arbitrary position, and an integral of the third kind with two logarithmic infinities, also of arbitrary position; the corresponding properties for p = 1 are proved above.There is, however, a difference of essential kind in regard to the inversion of integrals of the first kind; if u = ∫R(x, y)dx be such an integral, it can be shown, in common with all algebraic integrals associated with ƒ(x, y) = 0, to have 2p linearly independent additive constants of indeterminateness; the upper limit of the integral cannot therefore, as we have shown, be a single valued function of the value of the integral. The corresponding theorem, if ∫Ri(x, y)dx denote one of the integrals of the first kind, is that the p equations∫ Ri(x1, y1)dx1+ ... + ∫ Ri(xp, yp)dxp= ui,determine the rational symmetric functions of the p positions (x1, y1), ... (xp, yp) as single valued functions of the p variables, u1, ... up. It is thus necessary to enter into the theory of functions of several independent variables; and the equation ƒ(x, y) = 0 is thus not, in this way, capable of solution by single valued functions of one variable. That solution in fact is to be sought with the help of automorphic functions, which, however, as has been remarked, have, for p > 1, an infinite number of essential singularities.
The algebraical integrals ∫ R(x, y)dx associated with this may as before be divided into those of thefirst kind, which have no infinities, those of thesecond kind, possessing only algebraical infinities, and those of thethird kind, for which logarithmic infinities enter. Here there is a certain number, p, greater than unity, of linearly independent integrals of the first kind; and this number p is unaltered by any birational transformation of the fundamental equation ƒ(x, y) = 0; a rational function can be constructed with poles of the first order at p + 1 arbitrary positions (x, y), satisfying ƒ(x, y) = 0, but not with a fewer number unless their positions are chosen properly, a property we found for the case p = 1; and p is the number of linearly independent curves of order n − 3 passing through the double points of the curve of order n expressed by ƒ(x, y) = 0. Again any integral of the second kind can be expressed as a sum of p integrals of this kind, with poles of the first order at arbitrary positions, together with rational functions and integrals of the first kind; and an integral of the second kind can be found with one pole of the first order of arbitrary position, and an integral of the third kind with two logarithmic infinities, also of arbitrary position; the corresponding properties for p = 1 are proved above.
There is, however, a difference of essential kind in regard to the inversion of integrals of the first kind; if u = ∫R(x, y)dx be such an integral, it can be shown, in common with all algebraic integrals associated with ƒ(x, y) = 0, to have 2p linearly independent additive constants of indeterminateness; the upper limit of the integral cannot therefore, as we have shown, be a single valued function of the value of the integral. The corresponding theorem, if ∫Ri(x, y)dx denote one of the integrals of the first kind, is that the p equations
∫ Ri(x1, y1)dx1+ ... + ∫ Ri(xp, yp)dxp= ui,
determine the rational symmetric functions of the p positions (x1, y1), ... (xp, yp) as single valued functions of the p variables, u1, ... up. It is thus necessary to enter into the theory of functions of several independent variables; and the equation ƒ(x, y) = 0 is thus not, in this way, capable of solution by single valued functions of one variable. That solution in fact is to be sought with the help of automorphic functions, which, however, as has been remarked, have, for p > 1, an infinite number of essential singularities.
§ 25.Monogenic Functions of Several Independent Variables.—A monogenic function of several independent complex variables ui, ... upis to be regarded as given by an aggregate of power series all obtainable by continuation from any one of them in a manner analogous to that before explained in the case of one independent variable. The singular points, defined as the limiting points of the range over which such continuation is possible, may either bepoles, orpolar points of indetermination, oressential singularities.
A pole is a point (u(0)1, ... u(0)p) in the neighbourhood of which the function is expressible as a quotient of converging power series in u1− u(0)1... up− u(0)p; of these the denominator series D must vanish at (u(0)1, ... u(0)p), since else the fraction is expressible as a power series and the point is not a singular point, but the numerator series N must not also vanish at (u(0)1, ... u(0)p), or if it does, it must be possible to write D = MD0, N = MN0, where M is a converging power series vanishing at (u(0)1, ...u(0)p), and N0is a converging power series, in (u1− u(0)1... up− u(0)p), not so vanishing. A polar point of indetermination is a point about which the function can be expressed as a quotient of two converging power series, both of which vanish at the point. As in such a simple case as (Ax + By)/ (ax + by), about x = 0, y = 0, it can be proved that then the function can be made to approach to any arbitrarily assigned value by making the variables u1, ... upapproach to u(0)1, ... u(0)pby a proper path. It is the necessary existence of such polar points of indetermination, which in case p > 2 are not merely isolated points, which renders the theory essentially more difficult than that of functions of one variable. An essential singularity is any which does not come under one of the two former descriptions and includes very various possibilities. A point at infinity in this theory is one for which any one of the variables u1, ... upis indefinitely great; such points are brought under the preceding definitions by meansof the convention that for u(0)i= ∞, the difference ui− u(0)iis to be understood to stand for u−1i. This being so, a single valued function of u1, ... upwithout essential singularities for infinite or finite values of the variables can be shown, by induction, to be, as in the case of p = 1, necessarily a rational function of the variables. A function having no singularities for finite values of all the variables is as before called an integral function; it is expressible by a power series converging for all finite values of the variables; a single valued function having for finite values of the variables no singularities other than poles or polar points of indetermination is called a meromorphic function; as for p = 1 such a function can be expressed as a quotient of two integral functions having no common zero point other than the points of indetermination of the function; but the proof of this theorem is difficult.The single valued functions which occur, as explained above, in the inversion of algebraic integrals of the first kind, for p > 1, are meromorphic. They must also be periodic, unaffected that is when the variables u1, ... uparesimultaneouslyincreased each by a proper constant, these being the additive constants of indeterminateness for the p integrals ∫ Ri(x, y)dx arising when (x, y) makes a closed circuit, the same for each integral. The theory of such single valued meromorphic periodic functions is simpler than that of meromorphic functions of several variables in general, as it is sufficient to consider only finite values of the variables; it is the natural extension of the theory of doubly periodic functions previously discussed. It can be shown to reduce, though the proof of this requires considerable developments of which we cannot speak, to the theory of a single integral function of u1, ... up, called theTheta Function. This is expressible as a series of positive and negative integral powers of quantities exp (c1u1), exp (c2u2), ... exp (cpup), wherein c1, ... cpare proper constants; for p = 1 this theta function is essentially the same as that above given under a different form (see § 14,Doubly Periodic Functions), the function σ(u). In the case of p = 1, all meromorphic functions periodic with the same two periods have been shown to be rational functions of two of them connected by a single algebraic equation; in the same way all meromorphic functions of p variables, periodic with the same sets of simultaneous periods, 2p sets in all, can be shown to be expressible rationally in terms of p + 1 such periodic functions connected by a single algebraic equation. Let x1, ... xp, y denote p + 1 such functions; then each of the partial derivatives dxi/∂uiwill equally be a meromorphic function of the same periods, and so expressible rationally in terms of x1, ... xp, y; thus there will exist p equations of the formdxi= R1du1+ ... + Rpdup,and hence p equations of the formdui= Hi, 1dx1+ ... + Hi, pdxp,wherein Hi, jare rational functions of x1, ... xp, y, these being connected by a fundamental algebraic (rational) equation, say ƒ(x1, ... xp, y) = 0. This then is the generalized form of the corresponding equation for p = 1.
A pole is a point (u(0)1, ... u(0)p) in the neighbourhood of which the function is expressible as a quotient of converging power series in u1− u(0)1... up− u(0)p; of these the denominator series D must vanish at (u(0)1, ... u(0)p), since else the fraction is expressible as a power series and the point is not a singular point, but the numerator series N must not also vanish at (u(0)1, ... u(0)p), or if it does, it must be possible to write D = MD0, N = MN0, where M is a converging power series vanishing at (u(0)1, ...u(0)p), and N0is a converging power series, in (u1− u(0)1... up− u(0)p), not so vanishing. A polar point of indetermination is a point about which the function can be expressed as a quotient of two converging power series, both of which vanish at the point. As in such a simple case as (Ax + By)/ (ax + by), about x = 0, y = 0, it can be proved that then the function can be made to approach to any arbitrarily assigned value by making the variables u1, ... upapproach to u(0)1, ... u(0)pby a proper path. It is the necessary existence of such polar points of indetermination, which in case p > 2 are not merely isolated points, which renders the theory essentially more difficult than that of functions of one variable. An essential singularity is any which does not come under one of the two former descriptions and includes very various possibilities. A point at infinity in this theory is one for which any one of the variables u1, ... upis indefinitely great; such points are brought under the preceding definitions by meansof the convention that for u(0)i= ∞, the difference ui− u(0)iis to be understood to stand for u−1i. This being so, a single valued function of u1, ... upwithout essential singularities for infinite or finite values of the variables can be shown, by induction, to be, as in the case of p = 1, necessarily a rational function of the variables. A function having no singularities for finite values of all the variables is as before called an integral function; it is expressible by a power series converging for all finite values of the variables; a single valued function having for finite values of the variables no singularities other than poles or polar points of indetermination is called a meromorphic function; as for p = 1 such a function can be expressed as a quotient of two integral functions having no common zero point other than the points of indetermination of the function; but the proof of this theorem is difficult.
The single valued functions which occur, as explained above, in the inversion of algebraic integrals of the first kind, for p > 1, are meromorphic. They must also be periodic, unaffected that is when the variables u1, ... uparesimultaneouslyincreased each by a proper constant, these being the additive constants of indeterminateness for the p integrals ∫ Ri(x, y)dx arising when (x, y) makes a closed circuit, the same for each integral. The theory of such single valued meromorphic periodic functions is simpler than that of meromorphic functions of several variables in general, as it is sufficient to consider only finite values of the variables; it is the natural extension of the theory of doubly periodic functions previously discussed. It can be shown to reduce, though the proof of this requires considerable developments of which we cannot speak, to the theory of a single integral function of u1, ... up, called theTheta Function. This is expressible as a series of positive and negative integral powers of quantities exp (c1u1), exp (c2u2), ... exp (cpup), wherein c1, ... cpare proper constants; for p = 1 this theta function is essentially the same as that above given under a different form (see § 14,Doubly Periodic Functions), the function σ(u). In the case of p = 1, all meromorphic functions periodic with the same two periods have been shown to be rational functions of two of them connected by a single algebraic equation; in the same way all meromorphic functions of p variables, periodic with the same sets of simultaneous periods, 2p sets in all, can be shown to be expressible rationally in terms of p + 1 such periodic functions connected by a single algebraic equation. Let x1, ... xp, y denote p + 1 such functions; then each of the partial derivatives dxi/∂uiwill equally be a meromorphic function of the same periods, and so expressible rationally in terms of x1, ... xp, y; thus there will exist p equations of the form
dxi= R1du1+ ... + Rpdup,
and hence p equations of the form
dui= Hi, 1dx1+ ... + Hi, pdxp,
wherein Hi, jare rational functions of x1, ... xp, y, these being connected by a fundamental algebraic (rational) equation, say ƒ(x1, ... xp, y) = 0. This then is the generalized form of the corresponding equation for p = 1.
§ 26.Multiply-Periodic Functions and the Theory of Surfaces.—The theory of algebraic integrals ∫ R(x, y)dx, wherein x, y are connected by a rational equation ƒ(x, y) = 0, has developed concurrently with the theory of algebraic curves; in particular the existence of the number p invariant by all birational transformations is one result of an extensive theory in which curves capable of birational correspondence are regarded as equivalent; this point of view has made possible a general theory of what might otherwise have remained a collection of isolated theorems.
In recent years developments have been made which point to a similar unity of conception as possible for surfaces, or indeed for algebraic constructs of any number of dimensions. These developments have been in two directions, at first followed independently, but now happily brought into the most intimate connexion. On the analytical side, E. Picard has considered the possibility of classifying integrals of the form ∫(Rds + Sdy), belonging to a surface ƒ(x, y, z) = 0, wherein R and S are rational functions of x, y, z, according as they are (1) everywhere finite, (2) have poles, which then lie along curves upon the surface, or (3) have logarithmic infinities, also then lying along curves, and has brought the theory to a high degree of perfection. On the geometrical side A. Clebsch and M. Noether, and more recently the Italian school, have considered the geometrical characteristics of a surface which are unaltered by birational transformation. It was first remarked that for surfaces of order n there are associated surfaces of order n − 4, having properties in relation thereto analogous to those of curves of order n − 3 for a plane curve of order n; if such a surface ƒ(x, y, z) = 0 have a double curve with triple points triple also for the surface, and φ(x, y, z) = 0 be a surface of order n − 4 passing through the double curve, the double integral∫ ∫φ dx dy∂f/∂zis everywhere finite; and, the most general everywhere finite integral of this form remains invariant in a birational transformation of the surface ƒ, the theorem being capable of generalization to algebraic constructs of any number of dimensions. The number of linearly independent surfaces of order n − 4, possessing the requisite particularity in regard to the singular lines and points of the surface, is thus a number invariant by birational transformation, and the equality of these numbers for two surfaces is a necessary condition of their being capable of such transformation. The number of surfaces of order m having the assigned particularity in regard to the singular points and lines of the fundamental surface can be given by a formula for a surface of given singularity; but the value of this formula for m = n − 4 is not in all cases equal to the actual number of surfaces of order n − 4 with the assigned particularity, and for a cone (or ruled surface) is in fact negative, being the negative of the deficiency of the plane section of the cone. Nevertheless this number for m = n − 4 is also found to be invariant for birational transformation. This number, now denoted by pa, is then a second invariant of birational transformation. The former number, of actual surfaces of order n − 4 with the assigned particularity in regard to the singularities of the surface, is now denoted by pg. The difference pg− pa, which is never negative, is a most important characteristic of a surface. When it is zero, as in the case of the general surface of order n, and in a vast number of other ordinary cases, the surface is called regular.On a plane algebraical curve we may consider linear series of sets of points, obtained by the intersection with it of curves λφ + λ1φ1+ ... = 0, wherein λ, λ1, ... are variable coefficients; such a series consists of the sets of points where a rational function of given poles, belonging to the construct ƒ(x, y) = 0, has constant values. And we may consider series of sets of points determined by variable curves whose coefficients are algebraical functions, not necessarily rational functions, of parameters. Similarly on a surface we may consider linear systems of curves, obtained by the intersection with the given surface of variable surfaces λφ + λ1φ1+ ... = 0, and may consider algebraic systems, of which the individual curve is given by variable surfaces whose coefficients are algebraical, not necessarily rational, functions of parameters. Of a linear series upon a plane curve there are two numbers manifestly invariant in birational transformation, theorder, which is the number of points forming a set of the series, and thedimension, which is the number of parameters λ1/λ, λ2/λ, ... entering linearly in the equation of the series. The series iscompletewhen it is not contained in a series of the same order but of higher dimension. So for a linear system of curves upon a surface, we have three invariants for birational transformation; theorder, being in the number of variable intersections of two curves of the system, thedimension, being the number of linear parameters λ1/λ, λ2/λ, ... in the equation for the system, and thedeficiencyof the individual curves of the system. Upon any curve of the linear system the other curves of the system define a linear series, called thecharacteristicseries; but even when the linear system is complete, that is, not contained in another linear system of the same order and higher dimension, it does not follow that the characteristic series is complete; it may be contained in a series whose dimension is greater by pg− pathan its own dimension. When this is so it can be shown that the linear system of curves is contained in an algebraic system whose dimension is greater by pg− pathan the dimension of the linear system. The extra p = pg− pavariable parameters so entering may be regarded as the independent co-ordinates of an algebraic construct ƒ(y, x1, ... xp) = 0; this construct has the property that its co-ordinates are single valued meromorphic functions of p variables, which are periodic, possessing 2p systems of periods; the p variables are expressible in the formsui= ∫ R1(x, y) dx1+ ... + Rp(x, y) dxp,wherein Ri(x, y) denotes a rational function of x1, ... xpand y. The original surface has correspondingly p integrals of the form ∫(R dx + S dy), wherein R, S are rational in x, y, z, which are everywhere finite; and it can be shown that it has no other such integrals. From this point of view, then, the number p, = pg− pais, for a surface, analogous to the deficiency of a plane curve; another analogy arises in the comparison of the theorems: for a plane curve of zero deficiency there exists no algebraic series of sets of points which does not consist of sets belonging to a linear series; for a surface for which pg− pa= 0 there exists no algebraic system of curves not contained in a linear system.But whereas for a plane curve of deficiency zero, the co-ordinates of the points of the curve are rational functions of a single parameter, it is not necessarily the case that for a surface having pg− pa= 0 the co-ordinates of the points are rational functions of two parameters; it is necessary that pg− pa= 0, but this is not sufficient. For surfaces, beside the pglinearly independent surfaces of order n − 4 having a definite particularity at the singularities of the surface, it is useful to consider surfaces of order k(n − 4), also having each a definite particularity at the singularities, the number of these, not containing the original surface as component, which are linearly independent, is denoted by Pk. It can then be stated that a sufficient condition for a surface to be rational consists of the two conditions pa= 0, P2= 0. More generally it becomes a problem to classify surfaces according to the values of the various numbers which are invariant under birational transformation, and to determine for each the simplest form of surface to which it is birationally equivalent. Thus, for example, the hyperelliptic surface discussed by Humbert,of which the co-ordinates are meromorphic functions of two variables of the simplest kind, with four sets of periods, is characterized by pg= 1, pa= −1; or again, any surface possessing a linear system of curves of which the order exceeds twice the deficiency of the individual curves diminished by two, is reducible by birational transformation to a ruled surface or is a rational surface. But beyond the general statement that much progress has already been made in this direction, of great interest to the student of the theory of functions, nothing further can be added here.Bibliography.—The learner will find a lucid introduction to the theory in E. Goursat,Cours d’analyse mathématique, t. ii. (Paris, 1905), or, with much greater detail, in A.R. Forsyth,Theory of Functions of a Complex Variable(2nd ed., Cambridge, 1900); for logical rigour in the more difficult theorems, he should consult W.F. Osgood,Lehrbuch der Functionentheorie, Bd. i. (Leipzig, 1906-1907); for greater precision in regard to the necessary quasi-geometrical axioms, beside the indications attempted here, he should consult W.H. Young,The Theory of Sets of Points(Cambridge, 1906), chs. viii.-xiii., and C. Jordan,Cours d’analyse, t. i. (Paris, 1893), chs. i., ii.; a comprehensive account of theTheory of Functions of Real Variablesis by E.W. Hobson (Cambridge, 1907). Of the theory regarded as based after Weierstrass upon the theory of power series, there is J. Harkness and F. Morley,Introduction to the Theory of Analytic Functions(London, 1898), an elementary treatise; for the theory of the convergence of series there is also T.J. I’A. Bromwich,An Introduction to the Theory of Infinite Series(London, 1908); but the student should consult the collected works of Weierstrass (Berlin, 1894 ff.), and the writings of Mittag-Leffler in the early volumes of theActa mathematica; earlier expositions of the theory of functions on the basis of power series are in C. Méray,Leçons nouvelles sur l’analyse infinitésimale(Paris, 1894), and in Lagrange’s books on the Theory of Functions. An account of the theory of potential in its applications to the present theory is found in most treatises; in particular consult E. Picard,Traité d’analyse, t. ii. (Paris, 1893). For elliptic functions there is an introductory book, P. Appell and E. Lacour,Principes de la théorie des fonctions elliptiques et applications(Paris, 1897), beside the treatises of G.H. Halphen,Traité des fonctions elliptiques et de leurs applications(three parts, Paris, 1886 ff.), and J. Tannery et J. Molk,Éléments de la théorie des fonctions elliptiques(Paris, 1893 ff.); a book, A.G. Greenhill,The Applications of Elliptic Functions(London, 1892), shows how the functions enter in problems of many kinds. For modular functions there is an extensive treatise, F. Klein and R. Fricke,Theorie der elliptischen Modulfunctionen(Leipzig, 1890); see also the most interesting smaller volume, F. Klein,Über das Ikosaeder(Leipzig, 1884) (also obtainable in English). For the theory of Riemann’s surface, and algebraic integrals, an interesting introduction is P. Appeil and E. Goursat,Théorie des fonctions algébriques et de leurs intégrales; for Abelian functions see also H. Stahl,Theorie der Abel’schen Functionen(Leipzig, 1896), and H.F. Baker,An Introduction to the Theory of Multiply Periodic Functions(Cambridge, 1907), and H.F. Baker,Abel’s Theorem and the Allied Theory, including the Theory of the Theta Functions(Cambridge, 1897); for theta functions of one variable a standard work is C.G. Jacobi,Fundamenta nova, &c.(Königsberg, 1828); for the general theory of theta functions, consult W. Wirtinger,Untersuchungen über Theta-Functionen(Leipzig, 1895). For a history of the theory of algebraic functions consult A. Brill and M. Noether,Die Entwicklung der Theorie der algebraischen Functionen in älterer und neuerer Zeit, Bericht der deutschen Mathematiker-Vereinigung(1894); and for a special theory of algebraic functions, K. Hensel and G. Landsberg,Theorie der algebraischen Function u.s.w.(Leipzig, 1902). The student will, of course, consult also Riemann’s and Weierstrass’sGes. Werke. For the applications to geometry in general an important contribution, of permanent value, is E. Picard and G. Simart,Théorie des fonctions algébriques de deux variables indépendantes(Paris, 1897-1906). This work contains, as Note v. t. ii. p. 485, a valuable summary by MM. Castelnuovo and Enriques,Sur quelques résultats nouveaux dans la théorie des surfaces algébriques, containing many references to the numerous memoirs to be found, for the most part, in the transactions of scientific societies and the mathematical journals of Italy.Beside the books above enumerated there exists an unlimited number of individual memoirs, often of permanent importance and only imperfectly, or too elaborately, reproduced in the pages of the volumes in which the student will find references to them. The GermanEncyclopaedia of Mathematics, and the Royal Society’sReference Catalogue of Current Scientific Literature, Pure Mathematics, published yearly, should also be consulted.
In recent years developments have been made which point to a similar unity of conception as possible for surfaces, or indeed for algebraic constructs of any number of dimensions. These developments have been in two directions, at first followed independently, but now happily brought into the most intimate connexion. On the analytical side, E. Picard has considered the possibility of classifying integrals of the form ∫(Rds + Sdy), belonging to a surface ƒ(x, y, z) = 0, wherein R and S are rational functions of x, y, z, according as they are (1) everywhere finite, (2) have poles, which then lie along curves upon the surface, or (3) have logarithmic infinities, also then lying along curves, and has brought the theory to a high degree of perfection. On the geometrical side A. Clebsch and M. Noether, and more recently the Italian school, have considered the geometrical characteristics of a surface which are unaltered by birational transformation. It was first remarked that for surfaces of order n there are associated surfaces of order n − 4, having properties in relation thereto analogous to those of curves of order n − 3 for a plane curve of order n; if such a surface ƒ(x, y, z) = 0 have a double curve with triple points triple also for the surface, and φ(x, y, z) = 0 be a surface of order n − 4 passing through the double curve, the double integral
is everywhere finite; and, the most general everywhere finite integral of this form remains invariant in a birational transformation of the surface ƒ, the theorem being capable of generalization to algebraic constructs of any number of dimensions. The number of linearly independent surfaces of order n − 4, possessing the requisite particularity in regard to the singular lines and points of the surface, is thus a number invariant by birational transformation, and the equality of these numbers for two surfaces is a necessary condition of their being capable of such transformation. The number of surfaces of order m having the assigned particularity in regard to the singular points and lines of the fundamental surface can be given by a formula for a surface of given singularity; but the value of this formula for m = n − 4 is not in all cases equal to the actual number of surfaces of order n − 4 with the assigned particularity, and for a cone (or ruled surface) is in fact negative, being the negative of the deficiency of the plane section of the cone. Nevertheless this number for m = n − 4 is also found to be invariant for birational transformation. This number, now denoted by pa, is then a second invariant of birational transformation. The former number, of actual surfaces of order n − 4 with the assigned particularity in regard to the singularities of the surface, is now denoted by pg. The difference pg− pa, which is never negative, is a most important characteristic of a surface. When it is zero, as in the case of the general surface of order n, and in a vast number of other ordinary cases, the surface is called regular.
On a plane algebraical curve we may consider linear series of sets of points, obtained by the intersection with it of curves λφ + λ1φ1+ ... = 0, wherein λ, λ1, ... are variable coefficients; such a series consists of the sets of points where a rational function of given poles, belonging to the construct ƒ(x, y) = 0, has constant values. And we may consider series of sets of points determined by variable curves whose coefficients are algebraical functions, not necessarily rational functions, of parameters. Similarly on a surface we may consider linear systems of curves, obtained by the intersection with the given surface of variable surfaces λφ + λ1φ1+ ... = 0, and may consider algebraic systems, of which the individual curve is given by variable surfaces whose coefficients are algebraical, not necessarily rational, functions of parameters. Of a linear series upon a plane curve there are two numbers manifestly invariant in birational transformation, theorder, which is the number of points forming a set of the series, and thedimension, which is the number of parameters λ1/λ, λ2/λ, ... entering linearly in the equation of the series. The series iscompletewhen it is not contained in a series of the same order but of higher dimension. So for a linear system of curves upon a surface, we have three invariants for birational transformation; theorder, being in the number of variable intersections of two curves of the system, thedimension, being the number of linear parameters λ1/λ, λ2/λ, ... in the equation for the system, and thedeficiencyof the individual curves of the system. Upon any curve of the linear system the other curves of the system define a linear series, called thecharacteristicseries; but even when the linear system is complete, that is, not contained in another linear system of the same order and higher dimension, it does not follow that the characteristic series is complete; it may be contained in a series whose dimension is greater by pg− pathan its own dimension. When this is so it can be shown that the linear system of curves is contained in an algebraic system whose dimension is greater by pg− pathan the dimension of the linear system. The extra p = pg− pavariable parameters so entering may be regarded as the independent co-ordinates of an algebraic construct ƒ(y, x1, ... xp) = 0; this construct has the property that its co-ordinates are single valued meromorphic functions of p variables, which are periodic, possessing 2p systems of periods; the p variables are expressible in the forms
ui= ∫ R1(x, y) dx1+ ... + Rp(x, y) dxp,
wherein Ri(x, y) denotes a rational function of x1, ... xpand y. The original surface has correspondingly p integrals of the form ∫(R dx + S dy), wherein R, S are rational in x, y, z, which are everywhere finite; and it can be shown that it has no other such integrals. From this point of view, then, the number p, = pg− pais, for a surface, analogous to the deficiency of a plane curve; another analogy arises in the comparison of the theorems: for a plane curve of zero deficiency there exists no algebraic series of sets of points which does not consist of sets belonging to a linear series; for a surface for which pg− pa= 0 there exists no algebraic system of curves not contained in a linear system.
But whereas for a plane curve of deficiency zero, the co-ordinates of the points of the curve are rational functions of a single parameter, it is not necessarily the case that for a surface having pg− pa= 0 the co-ordinates of the points are rational functions of two parameters; it is necessary that pg− pa= 0, but this is not sufficient. For surfaces, beside the pglinearly independent surfaces of order n − 4 having a definite particularity at the singularities of the surface, it is useful to consider surfaces of order k(n − 4), also having each a definite particularity at the singularities, the number of these, not containing the original surface as component, which are linearly independent, is denoted by Pk. It can then be stated that a sufficient condition for a surface to be rational consists of the two conditions pa= 0, P2= 0. More generally it becomes a problem to classify surfaces according to the values of the various numbers which are invariant under birational transformation, and to determine for each the simplest form of surface to which it is birationally equivalent. Thus, for example, the hyperelliptic surface discussed by Humbert,of which the co-ordinates are meromorphic functions of two variables of the simplest kind, with four sets of periods, is characterized by pg= 1, pa= −1; or again, any surface possessing a linear system of curves of which the order exceeds twice the deficiency of the individual curves diminished by two, is reducible by birational transformation to a ruled surface or is a rational surface. But beyond the general statement that much progress has already been made in this direction, of great interest to the student of the theory of functions, nothing further can be added here.
Bibliography.—The learner will find a lucid introduction to the theory in E. Goursat,Cours d’analyse mathématique, t. ii. (Paris, 1905), or, with much greater detail, in A.R. Forsyth,Theory of Functions of a Complex Variable(2nd ed., Cambridge, 1900); for logical rigour in the more difficult theorems, he should consult W.F. Osgood,Lehrbuch der Functionentheorie, Bd. i. (Leipzig, 1906-1907); for greater precision in regard to the necessary quasi-geometrical axioms, beside the indications attempted here, he should consult W.H. Young,The Theory of Sets of Points(Cambridge, 1906), chs. viii.-xiii., and C. Jordan,Cours d’analyse, t. i. (Paris, 1893), chs. i., ii.; a comprehensive account of theTheory of Functions of Real Variablesis by E.W. Hobson (Cambridge, 1907). Of the theory regarded as based after Weierstrass upon the theory of power series, there is J. Harkness and F. Morley,Introduction to the Theory of Analytic Functions(London, 1898), an elementary treatise; for the theory of the convergence of series there is also T.J. I’A. Bromwich,An Introduction to the Theory of Infinite Series(London, 1908); but the student should consult the collected works of Weierstrass (Berlin, 1894 ff.), and the writings of Mittag-Leffler in the early volumes of theActa mathematica; earlier expositions of the theory of functions on the basis of power series are in C. Méray,Leçons nouvelles sur l’analyse infinitésimale(Paris, 1894), and in Lagrange’s books on the Theory of Functions. An account of the theory of potential in its applications to the present theory is found in most treatises; in particular consult E. Picard,Traité d’analyse, t. ii. (Paris, 1893). For elliptic functions there is an introductory book, P. Appell and E. Lacour,Principes de la théorie des fonctions elliptiques et applications(Paris, 1897), beside the treatises of G.H. Halphen,Traité des fonctions elliptiques et de leurs applications(three parts, Paris, 1886 ff.), and J. Tannery et J. Molk,Éléments de la théorie des fonctions elliptiques(Paris, 1893 ff.); a book, A.G. Greenhill,The Applications of Elliptic Functions(London, 1892), shows how the functions enter in problems of many kinds. For modular functions there is an extensive treatise, F. Klein and R. Fricke,Theorie der elliptischen Modulfunctionen(Leipzig, 1890); see also the most interesting smaller volume, F. Klein,Über das Ikosaeder(Leipzig, 1884) (also obtainable in English). For the theory of Riemann’s surface, and algebraic integrals, an interesting introduction is P. Appeil and E. Goursat,Théorie des fonctions algébriques et de leurs intégrales; for Abelian functions see also H. Stahl,Theorie der Abel’schen Functionen(Leipzig, 1896), and H.F. Baker,An Introduction to the Theory of Multiply Periodic Functions(Cambridge, 1907), and H.F. Baker,Abel’s Theorem and the Allied Theory, including the Theory of the Theta Functions(Cambridge, 1897); for theta functions of one variable a standard work is C.G. Jacobi,Fundamenta nova, &c.(Königsberg, 1828); for the general theory of theta functions, consult W. Wirtinger,Untersuchungen über Theta-Functionen(Leipzig, 1895). For a history of the theory of algebraic functions consult A. Brill and M. Noether,Die Entwicklung der Theorie der algebraischen Functionen in älterer und neuerer Zeit, Bericht der deutschen Mathematiker-Vereinigung(1894); and for a special theory of algebraic functions, K. Hensel and G. Landsberg,Theorie der algebraischen Function u.s.w.(Leipzig, 1902). The student will, of course, consult also Riemann’s and Weierstrass’sGes. Werke. For the applications to geometry in general an important contribution, of permanent value, is E. Picard and G. Simart,Théorie des fonctions algébriques de deux variables indépendantes(Paris, 1897-1906). This work contains, as Note v. t. ii. p. 485, a valuable summary by MM. Castelnuovo and Enriques,Sur quelques résultats nouveaux dans la théorie des surfaces algébriques, containing many references to the numerous memoirs to be found, for the most part, in the transactions of scientific societies and the mathematical journals of Italy.
Beside the books above enumerated there exists an unlimited number of individual memoirs, often of permanent importance and only imperfectly, or too elaborately, reproduced in the pages of the volumes in which the student will find references to them. The GermanEncyclopaedia of Mathematics, and the Royal Society’sReference Catalogue of Current Scientific Literature, Pure Mathematics, published yearly, should also be consulted.