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