VOLUME XI SLICE III Frost to Fyzabad (continued 8)

there are three such constants, of the form A − B, A − C, A − D, which are not connected by any linear equation with integral coefficients, and z is not a single valued function of u.

§ 20. Elliptic Integrals.—An integral of the form ∫ R(z, s)dz, where s denotes the square root of a quartic polynomial in z, which may reduce to a cubic polynomial, and R denotes a rational function of z and s, is called an elliptic integral.

To each value of z belong two values of s, of opposite sign; starting, for some particular value of z, with a definite one of these two values, the sign to be attached to s for any other value of z will be determined by the path of integration for z. When z is in the neighbourhood of any finite value z0 for which the radical s is not zero, if we put z − z0 = t, we can find s − s0 = a power series in t, say s=s0 + Q(t); when z is in the neighbourhood of a value, a, for which s vanishes, if we put z = a + t², we shall obtain s = tQ(t), where Q(t) is a power series in t; when z is very large and s² is a quartic polynomial in z, if we put z−1 = t, we shall find s−1 = t²Q(t); when z is very large and s² is a cubic polynomial in z, if we put z−1 = t², we shall find s−l = t³Q(t). By means of substitutions of these forms the character of the integral ∫ R(z, s)dz may be investigated for any position of z; in any case it takes a form ∫ [Ht−m + Kt−m+1 + ... + Pt−1 + R + St + ... ]dt involving only a finite number of negative powers of t in the subject of integration. Consider first the particular case ∫ s−1dz; it is easily seen that neither for any finite nor for infinite values of z can negative powers of t enter; the integral is everywhere finite, and is said to be of the first kind; it can, moreover, be shown without difficulty that no integral ∫ R(z, s)dz, save a constant multiple of ∫ s−1dz, has this property. Consider next, s² being of the form a0z4 + 4a1z³ + ..., wherein a0 may be zero, the integral ∫ (a0z² + 2a1z) s−1dz; for any finite value of z this integral is easily proved to be everywhere finite; but for infinite values of z its value is of the form At−1 + Q(t), where Q(t) is a power series; denoting by √a0 a particular square root of a0 when a0 is not zero, the integral becomes infinite for z = ∞ for both signs of s, the value of A being + √a0 or − √a0 according as s is √a0·z² (1 + [2a1/a0] z−1 + ... ) or is the negative of this; hence the integral J1 = ∫ ( [a0z² + 2a1z]/s + √a0) dz becomes infinite when z is infinite, for the former sign of s, its infinite term being 2√a0·t−1 or 2a0·z, but does not become infinite for z infinite for the other sign of s. When a0 = 0 the signs of s for z = ∞ are not separated, being obtained one from the other by a circuit of z about an infinitely large circle, and the form obtained represents an integral becoming infinite as before for z = ∞, its infinite part being 2√a1·t−1 or 2√a1·√z. Similarly if z0 be any finite value of z which is not a root of the polynomial ƒ(z) to which s² is equal, and s0 denotes a particular one of the determinations of s for z=z0, the integral

J2 = ∫ { s²0 + ½(z − z0) ƒ′(z0) + s0 } dz,

(z − z0)² s (z − z0)²

wherein ƒ′(z) = dƒ(z)/dz, becomes infinite for z = z0, s = s0, but not for z = z0, s = −s0. its infinite term in the former case being the negative of 2s0(z − z0). For no other finite or infinite value of z is the integral infinite. If z = θ be a root of ƒ(z), in which case the corresponding value of s is zero, the integral

J3 = ½ƒ′(θ) ∫ dz

(z − θ) s

becomes infinite for z=0, its infinite part being, if z − θ = t², equal to −[ƒ′(θ)]½ t−1: and this integral is not elsewhere infinite. In each of these cases, of the integrals J1, J2, J3, the subject of integration has been chosen so that when the integral is written near its point of infinity in the form ∫[At−2 + Bt−1 + Q(t)] dt, the coefficient B is zero, so that the infinity is of algebraic kind, and so that, when there are two signs distinguishable for the critical value of z, the integral becomes infinite for only one of these. An integral having only algebraic infinities, for finite or infinite values of z, is called an integral of the second kind, and it appears that such an integral can be formed with only one such infinity, that is, for an infinity arising only for one particular, and arbitrary, pair of values (s, z) satisfying the equation s² = ƒ(z), this infinity being of the first order. A function having an algebraic infinity of the mth order (m > 1), only for one sign of s when these signs are separable, at (1) z = ∞, (2) z = z0, (3) z = a, is given respectively by (s d/dz)m−1 J1, (s d/dz)m−1 J2, (s d/dz)m−1 J3, as we easily see. If then we have any elliptic integral having algebraic infinities we can, by subtraction from it of an appropriate sum of constant multiples of J1, J2, J3 and their differential coefficients just written down, obtain, as the result, an integral without algebraic infinities. But, in fact, if J, J1 denote any two of the three integrals J1, J2, J3, there exists an equation AJ + BJ′ + Cƒs−1dz = rational function of s, z, where A, B, C are properly chosen constants. For the rational function

s + s0 + z √a0

z − z0

is at once found to become infinite for (z0, s0), not for (z0, −s0), its infinite part for the first point being 2s/(z − z0), and to become infinite for z infinitely large, and one sign of s only when these are separable, its infinite part there being 2z √a0 or 2 √a1 √z when a0 = 0. It does not become infinite for any other pair (z, s) satisfying the relation s2 = ƒ(z); this is in accordance with the easily verified equation

s + s0 + z √a0 − J1 + J2 + (a0z02 + 2a1z0) ∫ dz = 0;

z − z0 s

and there exists the analogous equation

s + z √a0 − J1 + J3 + (a0θ2 + 2a1θ) ∫ dz .

z − θ s

Consider now the integral

P = ∫ ( s + s0 + z √a0 ) dz ;

z − z0 2s

this is at once found to be infinite, for finite values of z, only for (z0, s0), its infinite part being log (z − z0), and for z = ∞, for one sign of s only when these are separable, its infinite part being −log t, that is −log z when a0 ≠ 0, and −log (z1/2) when a0 = 0. And, if ƒ(θ) = 0, the integral

P1 = ∫ ( s + z √a0 ) dz

z − θ 2s

is infinite at z = θ, s = 0 with an infinite part log t, that is log (z − θ)1/2, is not infinite for any other finite value of z, and is infinite like P for z = ∞. An integral possessing such logarithmic infinities is said to be of the third kind.

Hence it appears that any elliptic integral, by subtraction from it of an appropriate sum formed with constant multiples of the integral J3 and the rational functions of the form (s d/dz)m−1 J1 with constant multiples of integrals such as P or P1, with constant multiples of the integral u = ∫s−1dz, and with rational functions, can be reduced to an integral H becoming infinite only for z = ∞, for one sign of s only when these are separable, its infinite part being of the form A log t, that is, A log z or A log (z1/2). Such an integral H = ∫R(z, s)dz does not exist, however, as we at once find by writing R(z, s) = P(z) + sQ(z), where P(z), Q(z) are rational functions of z, and examining the forms possible for these in order that the integral may have only the specified infinity. An analogous theorem holds for rational functions of z and s; there exists no rational function which is finite for finite values of z and is infinite only for z = ∞ for one sign of s and to the first order only; but there exists a rational function infinite in all to the first order for each of two or more pairs (z, s), however they may be situated, or infinite to the second order for an arbitrary pair (z, s); and any rational function may be formed by a sum of constant multiples of functions such as

s + s0 + z √a0 or s + z √a0

z − z0 z − θ

and their differential coefficients.

The consideration of elliptic integrals is therefore reducible to that of the three

u = ∫ dz , J = ∫ ( a0z2 + 2a1z + z √a0 ) dz, P = ∫ ( s + s0 + z √a0 ) dz

s s z − z0 2s

respectively of the first, second and third kind. Now the equation s2 = a0z4 + ... = a0 (z − θ) (z − φ) (z − ψ) (z − χ), by putting

y = 2s (z − θ)−2 [a0 (θ − φ) (θ − ψ) (θ − χ) ]−1/2

x = 1 + 1 ( 1 + 1 + 1 )

z − θ 3 θ − φ θ − ψ θ − χ

is at once reduced to the form y2 = 4x3 − g2x − g3 = 4(x − e1) (x − e2) (x − e3), say; and these equations enable us to express s and z rationally in terms of x and y. It is therefore sufficient to consider three elliptic integrals

u = ∫ dx , J = ∫ xdx , P = ∫ y + y0 dx .

y y x − x0 2y

Of these consider the first, putting

u = ∫ (∞)(x) dx ,

y

where the limits involve not only a value for x, but a definite sign for the radical y. When x is very large, if we put x−1 = t2, y−1 = 2t3 (1 − ¼ g2t4 − ¼ g3t6)−1/2, we have

u = ∫ t0 (1 + 1⁄8 g2t4 + ... ) dt = t + 1⁄40 g2t5 + ...,

whereby a definite power series in u, valid for sufficiently small value of u, is found for t, and hence a definite power series for x, of the form

x = u−2 + 1⁄20 g2u2 + ...

Let this expression be valid for 0 < |u| < R, and the function defined thereby, which has a pole of the second order for u=0, be denoted by φ(u). In the range in question it is single valued and satisfies the differential equation

[φ′(u)]2 = 4[φ(u)]3 − g2φ(u) − g3;

in terms of it we can write x = φ(u), y = − φ′(u), and, φ′(u) being an odd function, the sign attached to y in the original integral for x = ∞ is immaterial. Now for any two values u, v in the range in question consider the function

F(u, v) = ¼ [ φ′(u) − φ′(v) ]2 − φ(u) − φ(v);

φ(u) − φ(v)

it is at once seen, from the differential equation, to be such that ∂F/∂u = ∂F/∂v; it is therefore a function of u + v; supposing |u + v| < R we infer therefore, by putting v = 0, that

φ(u + v) = ¼ [ φ′(u) − φ′(v) ]2 − φ(u) − φ(v).

φ(u) − φ(v)

By repetition of this equation we infer that if u1, ... un be any arguments each of which is in absolute value less than R, whose sum is also in absolute value less than R, then φ(u1 + ... + un) is a rational function of the 2n functions φ(us), φ′(us); and hence, if |u| < R, that

φ(u) = H [ φ ( u ), φ′ ( u ) ],

n n

where H is some rational function of the arguments φ(u/n), φ′(u/n). In fact, however, so long as |u/n| < R, each of the functions φ(u/n), φ′(u/n) is single valued and without singularity save for the pole at u=0; and a rational function of single valued functions, each of which has no singularities other than poles in a certain region, is also a single valued function without singularities other than poles in this region. We infer, therefore, that the function of u expressed by H [φ(u/n), φ′(u/n)] is single valued and without singularities other than poles so long as |u| < nR; it agrees with φ(u) when |u| < R, and hence furnishes a continuation of this function over the extended range |u| < nR. Moreover, from the method of its derivation, it satisfies the differential equation [φ′(u)]2 = 4[φ(u)]3 − g2φ(u) − g3. This equation has therefore one solution which is a single valued monogenic function with no singularities other than poles for any finite part of the plane, having in particular for u = 0, a pole of the second order; and the method adopted for obtaining this near u=0 shows that the differential equation has no other such solution. This, however, is not the only solution which is a single valued meromorphic function, a the functions φ(u + α), wherein α is arbitrary, being such. Taking now any range of values of u, from u = 0, and putting for any value of u, x = φ(u), y = −φ′(u), so that y2=4x3-g2x-g3, we clearly have

u = ∫ (∞)(x, y) dx ;

y

conversely if x0 = φ(u0), y0 = −φ′(u0) and ξ, η be any values satisfying η2 = 4ξ2 − g2ξ − g3, which are sufficiently near respectively to x0, y0, while v is defined by

v − u0 = − ∫ (ξ, η)(x0, y0) dξ ,

η

then ξ, η are respectively φ(v) and −φ′(v); for this equation leads to an expansion for ξ − x0 in terms of v = u0 and only one such expansion, and this is obtained by the same work as would be necessary to expand φ(v) when v is near to u0; the function φ(u) can therefore be continued by the help of this equation, from v = u0, provided the lower limit of |ξ − x0| necessary for the expansions is not zero in the neighbourhood of any value (x0, y0). In fact the function φ(u) can have only a finite number of poles in any finite part of the plane of u; each of these can be surrounded by a small circle, and in the portion of the finite part of the plane of u which is outside these circles, the lower limit of the radii of convergence of the expansions of φ(u) is greater than zero; the same will therefore be the case for the lower limit of the radii |ξ − x0| necessary for the continuations spoken of above provided that the values of (ξ, η) considered do not lead to infinitely increasing values of v; there does not exist, however, any definite point (ξ0, η0) in the neighbourhood of which the integral ∫ (ξ, η)(x0, y0) dξ/η increases indefinitely, it is only by a path of infinite length that the integral can so increase. We infer therefore that if (ξ, η) be any point, where η2 = 4ξ3 − g2ξ − g3, and v be defined by

v = ∫ (∞)(ξ, η) dx ,

y

then ξ = φ(v) and η = −φ′(v). Thus this equation determines (ξ, η) without ambiguity. In particular the additive indeterminatenesses of the integral obtained by closed circuits of the point of integration are periods of the function φ(u); by considerations advanced above it appears that these periods are sums of integral multiples of two which may be taken to be

ω = 2 ∫ ∞e1 dx , ω′ = 2 ∫ ∞e3 dx ;

y y

these quantities cannot therefore have a real ratio, for else, being periods of a monogenic function, they would, as we have previously seen, be each integral multiples of another period; there would then be a closed path for (x, y), starting from an arbitrary point (x0, y0), other than one enclosing two of the points (e1, 0), (e2, 0), (e3, 0), (∞, ∞), which leads back to the initial point (x0, y0), which is impossible. On the whole, therefore, it appears that the function φ(u) agrees with the function ℜ(u) previously discussed, and the discussion of the elliptic integrals can be continued in the manner given under § 14, Doubly Periodic Functions.

§ 21. Modular Functions.—One result of the previous theory is the remarkable fact that if

ω = 2 ∫ ∞e1 dx , ω′ = 2 ∫ ∞e3 dx ;

y y

where y2 = 4(x − e1) (x − e2) (x − e3), then we have

e1 = (½ω)−2 + Σ′ {[(m + ½) ω + m′ω′]−2 − [mω + m′ω′]−2},

and a similar equation for e3, where the summation refers to all integer values of m and m′ other than the one pair m = 0, m′ = 0. This, with similar results, has led to the consideration of functions of the complex ratio ω′/ω.

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,

(1 − θ + θ2)3 = (1 − h + h2)3 ;

θ2(1 − θ)2 h2(1 − h)2

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,

z(τ) = 4⁄27 [ℜ2 (½ω) + ℜ (z(½ω) ℜ (½ω′) + ℜ2 (½ω′)]3

ℜ2 (½ω) ℜ2 (½ω′) [ℜ (½ω) + ℜ (½ω′]2

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 the theory 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)m exp [(3m2 + m) iπτ].

12 12

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 (1⁄12 iπ) 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 a Group; the function J(τ), unaltered by all these substitutions, is called a Modular Function. More generally any function unaltered by all the substitutions of a group of linear substitutions of its variable is called an Automorphic Function. A rational function, of its variable h, of this character, is the function (1 − h + h2)3 h−2(1 − h)−2 presenting 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

s″′ − 3 ( s″ )2 = 1 − α2 + 1 − β2 + α2 + β2 − γ2 − 1 ,

s′ 2 s′ 2(x − 1)2 2x2 2x (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 by

λ = 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.

§ 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 that an 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 form Σ ∞n=0 anτ′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

K = ∫ 10 dt , K′ = ∫ 10 dt ;

√[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 equation

x(1 − x) d2y + [γ − (α + β + 1)x] dy − αβγ = 0

dx2 dx

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 intersect ƒ 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 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

u = ∫ (x)(∞) ZdX − XdZ ,

ZY

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 + u3 is 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

dx = xδm + δn ,

y F′(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 + u3 is independent of the straight line considered; if in particular this become the inflexional tangent each of u1, u2, u3 vanishes. 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 the first kind, which have no infinities, those of the second kind, possessing only algebraical infinities, and those of the third 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, ... up is 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 be poles, or polar points of indetermination, or essential 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 N0 is 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, ... up approach to u(0)1, ... u(0)p by 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, ... up is indefinitely great; such points are brought under the preceding definitions by means of the convention that for u(0)i = ∞, the difference ui − u(0)i is to be understood to stand for u−1i. This being so, a single valued function of u1, ... up without 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, ... up are simultaneously increased 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 the Theta Function. This is expressible as a series of positive and negative integral powers of quantities exp (c1u1), exp (c2u2), ... exp (cpup), wherein c1, ... cp are 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/∂ui will 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, j are 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/∂z

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, the order, which is the number of points forming a set of the series, and the dimension, which is the number of parameters λ1/λ, λ2/λ, ... entering linearly in the equation of the series. The series is complete when 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; the order, being in the number of variable intersections of two curves of the system, the dimension, being the number of linear parameters λ1/λ, λ2/λ, ... in the equation for the system, and the deficiency of the individual curves of the system. Upon any curve of the linear system the other curves of the system define a linear series, called the characteristic series; 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 − pa than 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 − pa than the dimension of the linear system. The extra p = pg − pa variable 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, ... xp and 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 − pa is, 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 pg linearly 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 the Theory of Functions of Real Variables is 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 the Acta 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’s Ges. 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 German Encyclopaedia of Mathematics, and the Royal Society’s Reference Catalogue of Current Scientific Literature, Pure Mathematics, published yearly, should also be consulted.

(H. F. Ba.)

1 The word “function” (from Lat. fungi, to perform) has many uses, with the fundamental sense of an activity special or proper to an office, business or profession, or to an organ of an animal or plant, the definite work for which the organ is an apparatus. From the use of the word, as in the Italian funzione, for a ceremony of the Roman Church, “function” is often employed for a public ceremony of any kind, and loosely of a social entertainment or gathering.

FUNDY, BAY OF, an inlet of the North Atlantic, separating New Brunswick from Nova Scotia. It is 145 m. long and 48 m. wide at the mouth, but gradually narrows towards the head, where it divides into Chignecto Bay to the north, which subdivides into Shepody Bay and Cumberland Basin (the French Beaubassin), and Minas Channel, leading into Minas Basin, to the east and south. Off its western shore opens Passamaquoddy Bay, a magnificent sheet of deep water with good anchorage, receiving the waters of the St Croix river and forming part of the boundary between New Brunswick and the state of Maine, The Bay of Fundy is remarkable for the great rise and fall of the tide, which at the head of the bay has been known to reach 62 ft. In Passamaquoddy Bay the rise and fall is about 25 ft., which gradually increases toward the narrow upper reaches. At spring tides the water in the Bay of Fundy is 19 ft. higher than it is in Bay Verte, in Northumberland Strait, only 15 m. distant. Though the bay is deep, navigation is rendered dangerous by the violence and rapidity of the tide, and in summer by frequent fogs. At low tide, at such points as Moncton or Amherst, only an expanse of red mud can be seen, and the tide rushes in a bore or crest from 3 to 6 ft. in height. Large areas of fertile marshes are situated at the head of the bay, and the remains of a submerged forest show that the land has subsided in the latest geological period at least 40 ft. The bay receives the waters of the St Croix and St John rivers, and has numerous harbours, of which the chief are St Andrews (on Passamaquoddy Bay) and St John in New Brunswick, and Digby and Annapolis (on an inlet known as Annapolis Basin) in Nova Scotia. It was first explored by the Sieur de Monts (d. c. 1628) in 1604 and named by him La Baye Française.

FUNERAL RITES, the ceremonies associated with different methods of disposing of the dead. (See also Burial and Burial Acts; Cemetery; and Cremation.) In general we have little record, except in their tombs, of races which, in a past measured not merely by hundreds but by thousands of years, occupied the earth; and exploration of these often furnishes our only clue to the religions, opinions, customs, institutions and arts of long vanished societies. In the case of the great culture folks of antiquity, the Babylonians, Egyptians, Hindus, Persians, Greeks and Romans, we have, besides their monuments, the evidence of their literatures, and so can know nearly as much of their rites as we do of our own. The rites of modern savages not only help us to interpret prehistoric monuments, but explain peculiarities in our own rituals and in those of the culture folks of the past of which the significance was lost or buried under etiological myths. We must not then confine ourselves to the rites of a few leading races, neglecting their less fortunate brethren who have never achieved civilization. It is better to try to classify the rites of all races alike according as they embody certain leading conceptions of death, certain fears, hopes, beliefs entertained about the dead, about their future, and their relations with the living.

The main ideas, then, underlying funeral rites may roughly be enumerated as follows:

1. The pollution or taboo attaching to a corpse.

2. Mourning.

3. The continued life of the dead as evinced in the housing and equipment of the dead, in the furnishing of food for them, and in the orientation and posture assigned to the body.

4. Communion with the dead in a funeral feast and otherwise.

5. Sacrifice for the dead and expiation of their sins.

6. Death witchery.

7. Protection of the dead from ghouls.

8. Fear of ghosts.

1. A dead body is unclean, and the uncleanness extends to things and persons which touch it. Hence the Jewish law (Num. v. 2) enacted that “whoever is unclean by the dead shall be put outside the camp, that they defile not the camp in the midst whereof the Lord dwells.” Such persons were unclean until the even, and might not eat of the holy things unless they bathed their flesh in water. A high priest might on no account “go in to any dead body” (Lev. xxi. 11). Why a corpse is so widely tabooed is not certain; but it is natural to see one reason in the corruption which in warm climates soon sets in. The common experience that where one has died another is likely to do so may also have contributed, though, of course, there was no scientific idea of infection. The old Persian scriptures are full of this taboo. He who has touched a corpse is “powerless in mind, tongue and hand” (Zend Avesta in Sacred Books of the East, pt. i. p. 120), and the paralysis is inflicted by the innumerable drugs or evil spirits which invest a corpse. Fire and earth, being alike creations of the good and pure god Ahuramazda, a body must not be burned or buried; and so the ancient Persians and their descendants the Parsees build Dakmas or “towers of silence” on hill-tops far from human habitations. Inside these the corpses are laid on a flagged terrace which drains into a central pit. Twice a year the bones, picked clean by dogs and birds of prey, are collected in the pit, and when it is full another tower is built. In ancient times perhaps the bodies of the magi or priests alone were exposed at such expense; the common folk were covered with wax and laid in the earth, the wax saving the earth from pollution. In Rome and Greece the corpse was buried by night, lest it should pollute the sunlight; and a trough of water was set at the door of the house of death that men might purify themselves when they came out, before mixing in general society. Priests and magistrates in Rome might not meet or look on a corpse, for they were thereby rendered unclean and incapable of fulfilling their official duties without undergoing troublesome rites of purification. At a Roman funeral, when the remains had been laid in the tomb, all present were sprinkled with lustral water from a branch of olive or laurel called aspergillum; and when they had gone home they were asperged afresh and stepped over a fire. The house was also swept out with a broom, probably lest the ghost of the dead should be lying about the floor. Many races, to avoid pollution, destroy the house and property of the deceased. Thus the Navahos pull down the hut in which he died, leaving its ruins on the ground; but if it be an expensive hut, a shanty is extemporized alongside, into which the dying man is transferred before death. No one will use the timbers of a hut so ruined. A burial custom of the Solomon Islands, noted by R.H. Codrington (The Melanesians, p. 255), may be dictated by the same scruple. There “the mourners having hung up a dead man’s arms on his house make great lamentations; all remains afterwards untouched, the house goes to ruin, mantled, as time goes on, with the vines of the growing yams, a picturesque and indeed, perhaps, a touching sight; for these things are not set up that they may in a ghostly manner accompany their former owner.” H. Oldenberg (Religion des Veda, p. 426) describes how Hindus shave themselves and cut off their nails after a death, at the same time that they wash, renew the hearth fire, and furnish themselves with new vessels. For the hair and nails may harbour pollution, just as the medieval Greeks believed that evil spirits could lurk in a man’s beard (Leo Allatius, De opinionibus quorundam Graecorum). The dead man’s body is shorn and the nails cut for a kindred reason; for it must be purified as much as can be before it is burned as an offering on the pyre and before he enters on a new sphere of existence.

2. We are accustomed to regard mourning costume as primarily an outward sign of our grief. Originally, however, the special garb seems to have been intended to warn the general public that persons so attired were unclean. In ancient Rome mourners stayed at home and avoided all feasts and amusements; laying aside gold, purple and jewels, they wore black dresses called lugubria or even skins. They cut neither hair nor beard, nor lighted fire. Under the emperors women began to wear white. On the west coast of Africa negroes wear white, on the Gold Coast red. The Chinese wear hemp, which is cheap, for mourning dress must as a rule be destroyed when the season of grief is past to get rid of the taboo. Among the Aruntas of Australia the wives of a dead man smear themselves with white pipe-clay until the last ceremonies are finished, sometimes adding ashes—this not to conceal themselves from the ghost (which may partly be the aim of some mourning costumes), but to show the ghost that they are duly sorrowing for their loss. These widows must not talk except on their hands for a whole year. “Among the Maoris,” says Frazer (Golden Bough, i. 323), “anyone who had handled a corpse, helped to convey it to the grave, or touched a dead man’s bones; was cut off from all intercourse and almost all communication with mankind. He could not enter any house, or come into contact with any person or thing, without utterly bedevilling them. He might not even touch food with his hands, which had become so frightfully tabooed or unclean as to be quite useless. Food would be set for him on the ground, and he would then sit or kneel down, and, with his hands carefully held behind his back, would gnaw at it as best he could.” Often a degraded outcast was kept in a village to feed mourners. Such a taboo is strictly similar to those which surround a sacred chief or his property, a menstruous woman or a homicide, rendering them dangerous to themselves and to all who approach them.

3. Primitive folk cannot conceive of a man’s soul surviving apart from his body, nor of another life as differing from this, and the dead must continue to enjoy what they had here. Accordingly the Patagonians kill horses at the grave that the dead may ride to Alhuemapu, or country of the dead. After a year they collect a chief’s bones, arrange them, tie them together and dress them in his best garments with beads and feathers. Then they lay him with his weapons in a square pit, round which dead horses are placed set upright on their feet by stakes. As late as 1781 in Poland F. Casimir’s horse was slain and buried with him. In the Caucasus a Christian lady’s jewels are buried with her. The Hindus used to burn a man’s widow on his pyre, because he could not do without her; and St Boniface commends the self-sacrifice of the Wend widows who in his day burned themselves alive on their husbands’ pyres.

The tumuli met with all over the north of Europe (in the Orkneys alone 2000 remain) are regular houses of the dead, models of those they occupied in life. The greater the dignity of the deceased, the loftier was his barrow. Silbury hill is 170 ft. high; the tomb of Alyattes, father of Croesus, was a fourth of a league round; the Pyramids are still the largest buildings in existence; at Oberea in Tahiti is a barrow 267 ft. long, 87 wide and 44 high. Some Eskimo just leave a dead man’s body in his house, and shut it up, often leaving by his side a dog’s head to guide him on his last journey, along with his tools and kayak. The Sea Dyaks set a chief adrift in his war canoe with his weapons. So in Norse story Hake “was laid wounded on a ship with the dead men and arms; the ship was taken out to sea and set on fire.” The Viking was regularly buried in his ship or boat under a great mound. He sailed after death to Valhalla. In the ship was laid a stone as anchor and the tools, clothes, weapons and treasures of the dead. The Egyptians, whose land was the gift of the river Nile, equally believed that the dead crossed over water, and fashioned the hearse in the form of a boat. Hence perhaps was derived the Greek myth of Charon and the Styx, and the custom, which still survives in parts of Europe, of placing a coin in the mouth of the dead with which to pay the ferryman. The Egyptians placed in the tomb books of a kind to guide the dead to the next world. The Copts in a later age did the same, and to this custom we owe the recovery in Egypt of much ancient literature. The Armenians till lately buried with a priest his missal or gospel.

In Egyptian entombments of the XIIth to the XIVth dynasties were added above the sepulchres what Professor Petrie terms soul-houses, viz. small models of houses furnished with couch and table, &c., for the use of the ka or double whenever it might wish to come above ground and partake of meats and drinks. They recall, in point of size, the hut-urns of the Etruscans, but the latter had another use, for they contain incinerated remains. Etruscan tombs, like those of Egypt and Asia Minor, were made to resemble the dwelling-houses of the living, and furnished with coffered ceilings, panelled walls, couches, stools, easy chairs with footstools attached, all hewn out of the living rock (Dennis, Cities and Cemeteries of Etruria, vol i. p. lxx.).

Of the old Peruvian mummies in the Kircherian Museum at Rome, several are of women with babies in their arms, whence it is evident that a mother had her suckling buried with her; it would console her in the next world and could hardly survive her in this. The practice of burying ornaments, tools and weapons with the dead characterizes the inhumations of the Quaternary epoch, as if in that dim and remote age death was already regarded as the portal of another life closely resembling this. The cups, tools, weapons, ornaments and other articles deposited with the dead are often carefully broken or turned upside down and inside out; for the soul or manes of objects is liberated by such fracture or inversion and so passes into the dead man’s use and possession. For the same reason where the dead are burned, their properties are committed to the flames. The ghost of the warrior has a ghostly sword and buckler to fight with and a ghostly cup to drink from, and he is also nourished by the impalpable odour and reek of the animal victims sacrificed over his grave. Instead of valuable objects cheap images and models are often substituted; and why not, if the mere ghosts of the things are all that the wraith can enjoy? Thus Marco Polo (ii. 76) describes how in the land of Kinsay (Hang-chau) “the friends and relations make a great mourning for the deceased, and clothe themselves in hempen garments, and follow the corpse, playing on a variety of instruments and singing hymns to their idols. And when they come to the burning place they take representations of things cut out of parchment, such as caparisoned horses, male and female slaves, camels, armour, suits of cloth of gold (and money), in great quantities, and these things they put on the fire along with the corpse so that they are all burned with it. And they tell you that the dead man shall have all these slaves and animals of which the effigies are burned, alive in flesh and blood, and the money in gold, at his disposal in the next world; and that the instruments which they have caused to be played at his funeral, and the idol hymns that have been chaunted shall also be produced again to welcome him in the next world.” The manufacture of such paper simulacra for consumption at funerals is still an important industry in Chinese cities. The ancient Egyptians, assured that a man’s ka or double shall revivify his body, took pains to guard the flesh from corruption, steeping the corpse in natron and stuffing it with spices. A body so prepared is called a mummy (q.v.), and the custom was already of a hoary antiquity in 3200 B.C., when the oldest dated mummy we have was made. The bowels, removed in the process, were placed in jars over the corpse in the tomb, together with writing tablets, books, musical instruments, &c., of the dead. Cemeteries also remain full of mummies of crocodiles, cats, fish, cows and other sacred animals. The Greeks settled in Egypt learned to mummify their dead, but the custom was abhorrent to the Jews, although the Christian belief in the resurrection of the flesh must have been formed to a large extent under Egyptian influence. Half the superiority of the Jewish to other ancient religions lay in this, that it prescribed no funeral rites other than the simplest inhumation.

The dead all over the world and from remote antiquity have been laid not anyhow in the earth, but with the feet and face towards the region in which their future will be spent; the Samoans and Fijians towards the far west whither their souls have preceded them; the Guarayos with head turned eastwards because their god Tamoi has in that quarter “his happy hunting grounds where the dead will meet again” (Tylor, Prim. Cult. ii. 422). The legend is that Christ was buried with His head to the west, and the church follows the custom, more ancient than itself, of laying the dead looking to the East, because that is the attitude of prayer, and because at the last trump they will hurry eastwards. So in Eusebius (Hist. Eccl. 430.19) a martyr explains to his pagan judge that the heavenly Jerusalem, the fatherland of the pious, lay exactly in the east at the rising place of the sun. Where the body is laid out straight it is difficult to discern the presence of any other idea than that it is at rest. In Scandinavian barrows, e.g. in the one opened at Goldhavn in 1830, the skeletons have been found seated on a low stone bench round the wall of the grave chamber facing its opening, which always looks south or east, never north. Here the dead were continuing the drinking bouts they enjoyed on earth.

The Peruvians mummified their dead and placed them jointed and huddled up with knees to chin, looking toward the sunset, with the hands held before the face. In the oldest prehistoric tombs along the Nile the bodies are doubled up in the same position. It would seem as if in these and numerous other similar cases the dead were deliberately given in their graves the attitude of a foetus in the womb, and, as Dr Budge remarks (Egyptian Ideas of the Future Life, London, 1899, p. 162), “we may perhaps be justified in seeing in this custom the symbol of a hope that, as the child is born from this position into the world, so might the deceased be born into the life beyond the grave.” The late Quaternary skeletons of the Mentone cave were laid in a layer of ferrugineous earth specially laid down for them, and have contracted a red colour therefrom. Many other prehistoric skeletons found in Italy have a reddish colour, perhaps for the same reason, or because, as often to-day, the bones were stripped of flesh and painted. Ambrose relates that the skeletons of the martyrs Gervasius and Protasius, which he found and deposited A.D. 386 under the altar of his new basilica in Milan, were mirae magnitudinis ut prisca aetas ferebat, and were also coloured red. He imagined the red to be the remains of the martyrs’ blood! Hic sanguis clamat coloris indicio. Salomon Reinach has rightly divined that what Ambrose really hit upon was a prehistoric tomb. Red earth was probably chosen as a medium in which to lay a corpse because demons flee from red. Sacred trees and stones are painted red, and for the most solemn of their rites savages bedaub themselves with red clay. It is a favourite taboo colour.

4. A feast is an essential feature of every primitive funeral, and in the Irish “wake” it still survives. A dead man’s soul or double has to be fed at the tomb itself, perhaps to keep it from prowling about the homes of the survivors in search of victuals; and such food must also be supplied to the dead at stated intervals for months or years. Many races leave a narrow passage or tube open down to the cavity in which the corpse lies, and through it pour down drinks for the dead. Traces of such tubes are visible in the prehistoric tombs of the British Isles. However, such provision of food is not properly a funeral feast unless the survivors participate. In the Eastern churches and in Russia the departed are thus fed on the ninth, twelfth and fortieth days from death. “Ye appease the shades of the dead with wine and meals,” was the charge levelled at the Catholics by the 4th-century Manichaeans, and it has hardly ceased to be true even now after the lapse of sixteen centuries. The funeral feast proper, however, is either a meal of communion with or in the dead, which accompanies interment, or a banquet off the flesh of victims slain in atonement of the dead man’s sins. Some anthropologists see in the common meal held at the grave “the pledge and witness of the unity of the kin, the chief means, if not of making, at least of repairing and renewing it.”1 The flesh provided at these banquets is occasionally that of the dead man himself; Herodotus and Strabo in antiquity relate this of several half-civilized races in the East and West, and a similar story is told by Marco Polo of certain Tatars. Nor among modern savages are funeral feasts off the flesh of the dead unknown, and they seem to be intended to effect and renew a sacramental union or kinship of the living with the dead. The Uaupes in the Amazons incinerate a corpse a month after death, pound up the ashes, and mix them with their fermented drink. They believe that the virtues of the dead will thus be passed on to his survivors. The life of the tribe is kept inside the tribe and not lost. Such cannibal sacraments, however, are rare, and, except in a very few cases, the evidence for them weak. The slaying and eating of animal victims, however, at the tomb is universal and bears several meanings, separately or all at once. The animals may be slain in order that their ghosts may accompany the deceased in his new life. This significance we have already dwelt upon. Or it is believed that the shade feeds upon them, as the shades came up from Hades and lapped up out of a trench the blood of the animals slain by Ulysses. The survivors by eating the flesh of a victim, whose blood and soul the dead thus consume, sacramentally confirm the mystic tie of blood kinship with the dead. Or lastly, the victim may be offered for the sins of the dead. His sins are even supposed to be transferred into it and eaten by the priest. Such expiatory sacrifices of animals for the dead survive in the Christian churches of Armenia, Syria and of the East generally. Their vicarious character is emphasized in the prayers which accompany them, but the popular understanding of them probably combines all the meanings above enumerated. It has been suggested by Robertson Smith (Religion of the Semites, 336) that the world-wide customs of tearing the hair, rending the garments, and cutting and wounding the body were originally intended to establish a life-bond between the dead and the living. The survivors, he argues, in leaving portions of their hair and garments, and yet more by causing their own blood to stream over the corpse from self-inflicted wounds, by cutting off a finger and throwing it into the grave, leave what is eminently their own with the dead, so drawing closer their tie with him. Conversely, many savages daub themselves with the blood and other effluences of their dead kinsmen, and explain their custom by saying that in this way a portion of the dead is incorporated in themselves. Often the survivors, especially the widows, attach the bones or part of them to their persons and wear them, or at least keep them in their houses. The retention of the locks of the deceased and of parts of his dress is equally common. There is also another side to such customs. Having in their possession bits of the dead, and being so far in communion with him, the survivors are surer of his friendship. They have ensured themselves against ghosts who are apt to be by nature envious and mischievous. But whatever their original significance, the tearing of cheeks and hair and garments and cutting with knives are mostly expressions of real sorrow, and, as Robertson Smith remarks, of deprecation and supplication to an angry god or spirit. It must not be supposed that the savage or ancient man feels less than ourselves the poignancy of loss.

6. Death-witchery has close parallels in the witch and heretic hunts of the Christians, but, happily for us, only flourishes to-day among savages. Sixty % of the deaths which occur in West Africa are, according to Miss Mary Kingsley—a credible witness—believed to be due to witchcraft and sorcery. The blacks regard old age or effusion of blood as the sole legitimate causes of death. All ordinary diseases are in their opinion due to private magic on the part of neighbours, just as a widespread epidemic marks the active hatred “of some great outraged nature spirit, not of a mere human dabbler in devils.”2 Similarly in Christian countries an epidemic is set down to the wrath of a God offended by the presence of Jews, Arians and other heretics. The duty of an African witch-doctor is to find out who bewitched the deceased, just as it was of an inquisitor to discover the heretic. Every African post-mortem accordingly involves the murder of the person or persons who bewitched the dead man and caused him to die. The death-rate by these means is nearly doubled; but, since the use of poison against an obnoxious neighbour is common, the right person is occasionally executed. It is also well for neighbours not to quarrel, for, if they do and one of them dies of smallpox, the other is likely to be slain as a witch, and his lungs, liver and spleen impaled on a pole at the entrance of the village. It is the same case with the Australian blacks: “no such thing as natural death is realized by the native; a man who dies has of necessity been killed by some other man, or perhaps even by a woman, and sooner or later that man or woman will be attacked. In the normal condition of the tribe every death meant the killing of another individual.”3

7. Lastly, a primitive interment guards against the double risk of the ghost haunting the living and of ghouls or vampires taking possession of the corpse. The latter end is likely to be achieved if the body is cremated, for then there is no nidus to harbour the demon; but whether, in the remote antiquity to which belong many barrows containing incinerated remains, this motive worked, cannot be ascertained. The Indo-European race seems to have cremated at an early epoch, perhaps before the several races of East and West separated. In Christian funeral rites many prayers are for the protection of the body from violation by vampires, and it would seem as if such a motive dictated the architectural solidity of some ancient tombs. Christian graves were for protection regularly sealed with the cross; and the following is a characteristic prayer from the old Armenian rite for the burial of a layman:

“Preserve, Almighty Lord, this man’s spirit with all saints and with all lovers of Thy holy name. And do Thou seal and guard the sepulchre of Thy servant, Thou who shuttest up the depths and sealest them with Thy almighty right hand ... so let the seal of Thy Lordship abide unmoved upon this man’s dwelling-place and upon the shrine which guards Thy servant. And let not any filthy and unclean devil dare to approach him, such as assail the body and souls of the heathen, who possess not the birth of the holy font, and have not the dread seal laid upon their graves.”

A terrible and revolting picture of the superstitious belief in ghouls which violate Christian tombs is given by Leo Allatius (who held it) in his tract De opinionibus quorundam Graecorum (Paris, 1646). It was probably the fear of such demonic assaults on the dead that inspired the insanitary custom of burying the dead under the floors of churches, and as near as possible to the altar. In the Greek Church this practice was happily forbidden by the code of Justinian as well as by the older law in the case of churches consecrated with Encaenia and deposition of relics. In the Armenian Church the same rule holds, and Ephrem Syrus in his testament particularly forbade his body to be laid within a church. Such prohibitions, however, are a witness to the tendency in question.

The custom of lighting candles round a dead body and watching at its side all night was originally due to the belief that a corpse, like a person asleep, is specially liable to the assaults of demons. The practice of tolling a bell at death must have had a similar origin, for it was a common medieval belief that the sound of a consecrated bell drives off the demons which when a man dies gather near in the air to waylay his fleeting soul. For a like reason the consecrated bread of the Eucharist was often buried with believers, and St Basil is said to have specially consecrated a Host to be placed in his coffin.

8. Some of the rites described under the previous heads may be really inspired by the fear of the dead haunting the living, but it must be kept in mind that the taboo attaching to a dead body is one thing and fear of a ghost another. A corpse is buried or burned, or scaffolded on a tree, a tower or a house-top, in order to get it out of the way and shield society from the dangerous infection of its taboo; but ghosts quâ ghosts need not be feared and a kinsman’s ghost usually is not. On the contrary, it is fed and consoled with everything it needs, is asked not to go away but to stay, is in a thousand ways assured of the sorrow and sympathy of the survivors. Even if the body be eaten, it is merely to keep the soul of the deceased inside the circle of kinsmen, and Strabo asserts that the ancient Irish and Massagetae regarded it as a high honour to be so consumed by relatives. In Santa Cruz in Melanesia they keep the bones for arrow heads and store a skull in a box and set food before it “saying that this is the man himself” (R.H. Codrington, The Melanesians, p. 264), or the skull and jaw bone are kept and “are called mangite, which are saka, hot with spiritual power, and by means of which the help of the lio’a, the powerful ghost of the man whose relics these are, can be obtained” (ibid. p. 262). Here we have the savage analogue to Christian relics. So the Australian natives make pointing sticks out of the small bones of the arm, with which to bewitch enemies.