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3 ON THE RECENT PROGRESS OF ANALYSIS. 73 
__ the curve may be algebraical, it is necessary and sufficient, as M. Liouville 
__ has remarked, that the square of the modulus of the integral should be ra- 
_ tional, and less than unity. 
Ina subsequent memoir (Liouville’s Journal, x. 351) he has very much 
simplified the analytical part of his researches, and in the same Journal 
(x.421) has proved some remarkable properties of one class of what may be 
called elliptic curves. In the fourth number of the Cambridge and Dublin 
_ Mathematical Journal (p. 187), M. Serret has developed this part of the sub- 
ject, and has also given a general sketch of his previous papers. M. Liou- 
ville (Comptes Rendus, xxi. 1255, or his Journal, x. 456) has given a very 
elegant investigation of an analytical theorem established by M. Serret. 
In the fourteenth volume of Crelle’s Journal (p. 217), M. Gudermann has 
_ considered the rectification of the curve called the spherical ellipse, which is 
one of aclass of ¢urves formed by the intersection of a cone of the second 
_ order with a sphere. He has shown that its arcs represent an elliptic integral 
of the third kind. 
In the ninth volume of Liouville’s Journal (p. 155), Mr. W. Roberts proves 
_ that a cone of the second order, whose vertex lies on the surface of asphere, 
and one of whose external axes passes through the centre, intersects the 
sphere in a curve whose arcs will, according to circumstances, represent any 
» elliptic integral of the third kind and of the circular species ; or any elliptic 
integral of the same kind and of the logarithmic species, provided the angle 
of the modulus is less than half a right angle; or (subject to the same con- 
dition) any elliptic integral of the first kind ; or lastly, by a suitable modifi- 
cation, any elliptic integral of the second kind. The cases here excepted 
may be avoided by introducing known transformations. The cases in which 
_ the arcs represent elliptic integrals of the first kind, Mr. Roberts has pre- 
viously mentioned in the eighth volume of Liouville’s Journal (p.263). He 
has since given in the same Journal (x. 297), a general investigation of the 
subject, in which it is supposed that the vertex of the cone may have any 
position we please. M. Verhulst has represented the three kinds of elliptic 
_ integrals by means of sectorial areas of certain curves, and the function T by 
_ the volume of a certain solid. It is manifest, however, that it is incom- 
_ parably easier to do this than to represent these transcendents by means of 
_ the arcs of curves. 
_ Beside one or two other papers I may mention a tract by the Abbé Tor- 
_ tolini, on the geometrical representation of elliptic integrals of the second and 
third kinds. This tract, however, I have not seen. 
i Lagrange long since proved (vide Théorie des Fonctions Analytiques, 
p-85), that by means of a spherical triangle a geometrical representation of 
the addition of elliptic integrals of the first kind may easily be obtained, and 
_ that hence by a series of such triangles we are enabled to represent the mul- 
tiplication as well as the addition of these integrals. 
i M. Jacobi has given (Crelle’s Journal, iii. p. 376, or vide Liouville’s Journal, 
X. p. 435) a geometrical construction for the addition and multiplication of 
elliptic integrals of the first kind. It is founded on the properties of an irre- 
_ gular polygon inscribed in a circle, and the sides of which touch one or more 
other circles. It is to be remarked that Legendre, in giving an account of 
lis construction in one of the supplements to his last work, has only con- 
dered its application to multiplication and not to addition, and has been 
_ followed in this respect by M. Verhulst, whose treatise on elliptic functions 
_ has been already mentioned. In consequence of this, M. Chasles was led to 
believe that until the publication of his own researches, no construction for 
_ addition excepting that of Lagrange was known. But he has recently 
