By Serre J-P
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In his two notes Jacobi considered the transformation of elliptic functions, which he approached via the equation φ 0 dt 1 − c2 sin2 t ψ =m 0 dt 1 − k2 sin2 t . 28) Given an elliptic integral in terms of φ , it is required to transform it into another involving ψ . 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.
The reason seems to have been, as Krazer argued (1912), that on this occasion Euler remained too close to the geometry of conics and lemniscates with which he began. Nor did anyone else pick up the problem for a generation. As Enneper pointed out (Enneper 1876, 542) “the writings of Fagnano seem to have been very little known, and the writings of the great Euler too little read”. The man who first studied the lemniscatic integral from a purely functional point of view was Legendre. 13 In this paper he showed for the first time how any integral of the form Pdx R , where P is a rational function in x and R is the square root of a quartic (with real coefficients), can be simplified to one of the form Qdt (1 − t 2)(1 − c2t 2 ) .
Legendre was surprised to notice that he was the first to show that there were algebraic relations connecting the times of swings of a circular pendulum, just as there were for divisions of circular arcs. In Vol. II of his Exercises he dwelt at length on three problems: the rotation of a solid about a fixed point; the motion either in plane or space of a body attracted to two fixed bodies; and the attraction due to an homogeneous ellipsoid. In the first volume of his Trait´e he added four more examples: motion under central forces, the surface area of oblique cones, the surface area of ellipsoids, and the problem of 26 1 Elliptic Functions determining geodesics on an ellipsoid.