−4 ∫ 10 dz
(1 − z²)1/2

or 2π. Thus the additive indeterminateness of the integral is of the form 2kπ, where k is an integer, and the function z of u, which is sin (u), has 2π for period. Take now the case

u = ∫ (z)(z0) dz,
√{ (z − a) (z − b) (z − c) (z − d) }

adopting a definite determination for the phase of each of the factors z − a, z − b, z − c, z − d at the arbitrary point z0, and supposing the upper limit to refer, not only to a definite value of z, but also to a definite determination of the radical under the sign of integration. From z0 describe a closed loop about the point z = a, consisting, suppose, of a straight path from z0 to a, followed by a vanishing circle whose centre is at a, completed by the straight path from a to z0. Let similar loops be imagined for each of the points b, c, d, no two of these having a point in common. Let A denote the value obtained by the positive circuit of the first loop; this will be in fact equal to twice the integral taken from z0 along the straight path to a; for the contribution due to the vanishing circle is ultimately zero, and the effect of the circuit of this circle is to change the sign of the subject of integration. After the circuit about a, we arrive back at z0 with the subject of integration changed in sign; let B, C, D denote the values of the integral taken by the loops enclosing respectively b, c and d when in each case the initial determination of the subject of integration is that adopted in calculating A. If then we take a circuit from z0 enclosing both a and b but not either c or d, the value obtained will be A − B, and on returning to z0 the subject of integration will have its initial value. It appears thus that the integral is subject to an additive indeterminateness equal to any one of the six differences such as A − B. Of these there are only two linearly independent; for clearly only A − B, A − C, A − D are linearly independent, and in fact, as we see by taking a closed circuit enclosing all of a, b, c, d, we have A − B + C − D = 0; for there is no other point in the plane beside a, b, c, d about which the subject of integration suffers a change of sign, and a circuit enclosing all of a, b, c, d may by putting z = 1/ζ be reduced to a circuit about ζ = 0 about which the value of the integral is zero. The general value of the integral for any position of z and the associated sign of the radical, when we start with a definite determination of the subject of integration, is thus seen to be of the form u0 + m(A − B) + n(A − C), where m and n are integers. The value of A − B is independent of the position of z0, being obtainable by a single closed positive circuit about a and b only; it is thus equal to twice the integral taken once from a to b, with a proper initial determination of the radical under the sign of integration. Similar remarks to the above apply to any integral ∫ H(z)dz, in which H(z) is an algebraic function of z; in any such case H(z) is a rational function of z and a quantity s connected therewith by an irreducible rational algebraic equation ƒ(s, z) = 0. Such an integral ƒK(z, s)dz is called an Abelian Integral.

§ 19. Reversion of an Algebraic Integral.—In a limited number of cases the equation u = ∫ [z0 to z] H(z)dz, in which H(z) is an algebraic function of z, defines z as a single valued function of u. Several cases of this have been mentioned in the previous section; from what was previously proved under § 14, Doubly Periodic Functions, it appears that it is necessary for this that the integral should have at most two linearly independent additive constants of indeterminateness; for instance, for an integral

u = ∫ zz0 [(z − a) (z − b) (z − c) (z − d) (z − e) (z − f) ]−1/2dz,

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