XVIII. Researches on the geometrical properties of elliptic integrals

I. In placing before the Royal Society the following researches on the geometrical types of elliptic integrals, which nearly complete my investigations on this interesting subject, I may be permitted briefly to advert to what bad already been effected in this department of geometrical research. Legendre, to whom this important branch of mathematical science owes so much, devised a plane curve, whose rectification might be effected by an elliptic integral of the first order. Since that time many other geometers have followed his example, in contriving similar curves, to represent, either by their quadrature or rectification, elliptic functions. Of those who have been most successful in devising curves which should possess the required properties, may be mentioned M. Gudermann, M. Verhulst of Brussels, and M. Serret of Paris. These geometers however have succeeded in deriving from those curves scarcely any of the properties of elliptic integrals, even the most elementary. This barrenness in results was doubtless owing to the very artificial character of the genesis of those curves, devised, as they were, solely to satisfy one condition only of the general pro­blem. In 1841 a step was taken in the right direction. MM. Catalan and Gudermann, in the journals of Liouville and Crelle, showed how the arcs of spherical conic sec­tions might be represented by elliptic integrals of the third order and circular form. They did not, however, extend their investigations to the case of elliptic integrals of the third order and logarithmic form; nor even to that of the first order. These cases still remained, without any analogous geometrical representative, a blemish to the theory.