298 0^ THE RECENT PROGRESS OF ANALYSIS. 



as M. Liouville lias remarked, that the square of the modulus of 

 the integral should be rational, and less than unity. 



In a 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 Mathe- 

 matical Journal (p. 187), M. Serret has developed this part of 

 the subject, and has also given a general sketch of his pre- 

 vious papers. M. Liouville ( Comptes Eendus, xxi. 1 255, 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 a class of curves 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 a sphere, 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 inte- 

 gral of the third kind and of the circular species ; or any elliptic 

 integral of the same kind and of the logarithmic species, pro- 

 vided the angle of the modulus is less than half a right angle ; 

 or (subject to the same condition) any elliptic integral of the first 

 kind ; or lastly, by a suitable modification, 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 Eoberts has 

 previously 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 

 incomparably easier to do this than to represent these transcend- 

 ents by means of the arcs of curves. 



