| 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 + ...