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1852, 7). 36 1 Elliptic Functions Fig. 3 Carl Gustav Jacob Jacobi (From his Gesammelte Werke) Later, but richer, was Dirichlet’s comment, By June 1827 Jacobi had indeed come to some new ideas of his own. Specifically, he had found new ways to transform one elliptic integral into another by rational changes of the variable. He was the first to discover the existence of transformations of every degree. Not having seen Legendre’s Trait´e he did not know that Legendre had found a transformation of order 3.

Found, on setting the product only on the choice of c, not the angle φ ) that and = α (it is a constant depending F = (1 + co)(1 + coo) . . = α Φ . 18) This meant that F 1 (c) = π2 α and was easy to find from logarithm tables. When φ = π2 the limit Φ = π2 , and the corresponding value of F is α π2 . Moreover, the value of c tended rather rapidly to zero, so that the convergence was quite rapid. Legendre illustrated this with an example. 2460561. 19) from which is followed that it was often enough to use just the first two terms.

Each such transformation determines the values of the new modulus, k, and the number m. Jacobi observed that one can find such transformations by writing sin φ as a rational function of sin ψ thus: sin φ = U V , where U is a polynomial in odd powers of sin ψ up to the mth, and V is a polynomial in even powers of sin ψ up to the (m − 1)th. Such a transformation is said to be of order m. 3 Jacobi 37 this stage Jacobi only knew how to find the polynomials U and V when n = 3 or 5; the rest of his claim was strictly speaking only a conjecture.

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