300 ON THE RECENT PROGRESS OF ANALYSIS. 



on and off the arc which lies between its two extremities, the 

 ring will trace out a portion of an ellipse confocal to the former. 

 If for the first ellipse we substitute an hyperbola confocal with 

 the second, the sum of the arcs will be constant. From hence 

 a series of theorems is deduced, remarkable not only for their 

 elegance, but also for the facility with which they are obtained. 

 They furnish constructions for the addition and multiplication 

 of elliptic integrals. The whole of this investigation, of which 

 an account is given in the Comptes Rendus (Vol. xvn. p. 838, 

 and Vol. xix. p. 1239), shows, like others of M. Chasles, how 

 much is lost in treating geometrical questions by an exclusive 

 adherence to what may be called the method of co-ordination. 

 Invaluable as this method is, it yet often introduces considera- 

 tions foreign to the problem to which it is applied*. 



III. 



28. The first outline of a detailed theory of the higher 

 transcendents was given by Legendre in the third supplement to 

 his Trait& des Fonctions Elliptiques. He proposes to classify 

 the transcendents comprised in the general formula 



f /()*_ 



J (x a) V</># 



according to the degree of the polynomial (f>x, the first class being 

 that in which the index of this degree is three or four; the 

 second that in which it is five or six, and so on. The first class 

 therefore consists of elliptic integrals; all the others may be 

 designated as ultra-elliptic. This epithet, however, which was 

 proposed by Legendre, has not been so generally used as Jiyper- 

 elliptic, which was, I believe, first used by M. Jacobi. M. Jacobi, 

 however, has proposed to call the higher transcendents Abelian 

 integrals. 



The principle of Legendre's classification is to be found in 

 the mininum number of integrals to which the sum of any 



M. Chasles has also considered the subject of spherical conies, as well as that 

 of the lines of curvature and shortest lines on an ellipsoid. The latter has re. 

 cently engaged the attention of several distinguished mathematicians MM. Jacobi, 

 Joachimsthal, Liouville, Mac Cullagh and Roberts may be particularlv mentioned. 



