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X\'III. Researches on the Geometrical Properties of Elliptic Integrals, 
By the Rev. James Booth, LL.D., F.R.S. ^c. 
Received November 17, 1851, — Read January 22, 1852, 
Section I. 
1. 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 had 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 owdng to the very artificial character of the genesis of those 
curves, devised, as they were, solely to satisfy one condition only of the general pro- 
blem*. 
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 sec- 
tions 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. 
Some years ago, when engaged in the discussion of the problem of the rotation of 
a rigid body round a fixed point, by the help of an auxiliary ellipsoid, I had continu- 
ally brought under my notice, in the course of my investigations, the sections of a 
sphere by a concentric cone, or as they now are generally named, spherical conic 
* Legendre a cherche k repr^senter en general, la fonction dig. (c, par un arc de courbe ; mais ses ten- 
tatives ne nous ont pas semble heureuses, car il n’est parvenu ei resoudre completement le probleme, qu’en 
employant une courbe transcendante, dans laquelle I’amplitude f et I’arcs ont entre eux une relation geome- 
tnque encore plus difficile a saisir que dans la lemniscate. — Verhulst, Traite des Fonctions EUiptiques^ 
p. 295. 
2 s 
MDCCCLII. 
