DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 313 
rational transformation, make the equation U+/*V=0 represent a non-central sur- 
face instead of a central one, or vice versa. 
Although a remarkable relation exists between the areas and lengths of some of 
these hyperconics, such as the circle and the spherical ellipse, yet more distinctly to 
show the analogy which pervades all those curves, I have not had recourse in any 
case to the method of “elliptic quadratures,” as it is termed*. We cannot admit 
such a violation of the law of geometrical continuity as to suppose, that while a 
function in one state represents a curve line, in another, immediately succeeding, it 
must express an area. Such can only be taken as a conventional explanation, until 
the real one, characterized by the simplicity of truth, shall present itself. 
In the course of these investigations, it will be shown that the formulae for the 
comparison of elliptic integrals, which are given by Legendre and other writers on 
this subject, follow simply as geometrical inferences from the fundamental properties 
of those curves ; and that the ordinary conic sections are merely particular cases of 
those more general curves above referred to, under the name of hyperconic sections. 
It will doubtless appear not a little singular, that the principal properties of those 
functions, their classification, their transformations, the comparison of 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 func- 
tions. It is as if the formulse of trigonometry had been derived from an algebraical 
definition, before the geometrical conception of the circle had been admitted. 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 trigono- 
metry, or the theory of elliptic integrals, may best be interpreted as the development 
of the relations which exist between the arcs of hyperconic sections. 
Indeed it may with truth be asserted, that nearly all the principal functions, on 
which the resources of analysis have chiefly been exhausted, whether they be circular, 
logarithmic, exponential or elliptic, arise out of the solution of this one general pro- 
blem, to determine the length of an arc of a hyperconic section. 
It may be said, we cannot by this method derive any properties of elliptic inte- 
grals which may not algebraically be deduced from the fundamental expressions 
appropriately assumed. But surely no one will assert that the properties of curve 
lines should be algebraically developed, without any reference to their geometrical 
types. 
We might from algebraical expressions suitably chosen, derive every known property 
of curve lines, without having in any instance a conception of the geometrical types 
* En considerant les fonctions elliptiques comme des secteurs, dont Tangle est prdcisdment egal &, Tampli- 
tude (p, nous avons en Tavantage de justifier la denomination d’amplitude appliquee a Tangle (p ; et meme celle 
de fonctions elliptiques, en gdn^ral, puisque les courbes algdbraiques par lesquelles nous avons representes ces 
transcendantes, se construisent avec facilite an moyen des rayons vecteurs d’une ou de deux ellipses donnees. 
— Verhulst, Traite des Fonctions Elliptiques, p. 295. 
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