144 



second order represented an arc of a plane ellipse, was evident from 

 the beginning. Hence indeed the name " elliptic functions," de- 

 rived from a part, was given to the whole. Here then the question 

 naturally arose : What geometrical types did the first and third 

 orders represent } This question long remained without complete 

 solution ; and investigators in this department of analysis were 

 compelled to take the fundamental expressions as arbitrary data, 

 and to forgo the inquiry what the geometrical theorems were 

 v/llich these algebraical expressions represented. Various but un- 

 successful attempts were made by geometers to represent them by 

 qii£idratures, or by plane curves, either algebraical or transcendental. 

 About ten years ago, however, Messrs. Guderman and Catalan 

 showed that the circular form of the third order represented the 

 curve of intersection of a cone of the second degree and a concentric 

 sphere ; but they did not extend their researches to the first order, 

 nor to the logarithmic form of the third. 



The main object of the paper is to prove that elliptic integrals 

 of every kind, the parameter taking any value whatever between 

 positive and negative infinity, represent the intersections of surfaces 

 of the second order. 



These surfaces divide themselves into two classes, of which the 

 sphere and the paraboloid are the respective types ; from the former 

 arise the circular functions of the third order, from the other the 

 logarithmic and exponential. In the course of these investigations 

 it is shown that the formulae for the comparison of elliptic integrals, 

 which are given by Legendre, follow simply as geometrical inferences 

 from the fundamental properties of those curves. The ordinary 

 conic sections are merely particular cases of those more general 

 curves, to which the author has given the name Hyperconic Sections. 



The author remarks, that it will doubtless appear not a little sin- 

 gular, that the principal properties of these functions, their classi- 

 fication, their transformations, the comparison of elliptic integrals of 

 the third order, with conjugate or reciprocal parameters, were all 

 investigated and developed before geometers had any idea of the true 

 geometrical origin of those functions. It is as if the formulae of 

 common trigonometry had been derived from an algebraical defini- 

 tion, before the geometrical conception of the circle had been ad- 

 mitted. As trigonometry may be defined, the development of the 

 properties of circular arcs, whether described on a plane, or on the 

 surface of a sphere, so this higher trigonometry, or the theory of 

 elliptic integrals, may be defined as the development of the rela- 

 tions which exist between the arcs of hyperconic sections. 



It m^ty be said, we cannot by this method derive any properties 

 of elliptic integrals which may not algebraically be deduced from the 

 fundamental expressions appropriately assumed. It cannot, how- 

 ever, be truly asserted that the properties of curve lines should be 

 developed without any reference to their geometrical types. We 

 might, starting from certain algebraical expressions, derive every 

 known property of curve lines, without having in any instance a 

 conception of the geometrical types which they represent. The 



