252 ON THE RECENT PROGRESS OF ANALYSIS. 



and the second side of this equation is of course rational and 

 integrable. But the form of the f unction f(xv) is unnecessarily 

 restricted in order that this kind of reduction may be possible. 

 Nevertheless, Mr Talbot's papers, from their fulness of illus- 

 tration and the clear manner in which particular cases of the 

 general theory are worked out by independent methods, will be 

 found very useful in facilitating our conceptions of the branch 

 of analysis which forms as it were the link between the theory 

 of equations and the integral calculus. 



In Mr Talbot's second memoir (Phil. Trans. 1837, part 2. 

 p. 1) he has applied his method to certain geometrical theorems. 

 Three of them relate to the ellipse, and are proved by the three 

 following assumptions : 



vx > or = 7=- > or 



1-0* 



These assumptions are all cases of the following : 



where a, a 1 , c, c 1 are arbitrary quantities. The results of this 

 assumption are completely worked out by Legendre (ThSorie 

 des Fonctions Elliptiques, in. p. 192) in showing how the 

 known formulae of elliptic functions may be derived from Abel's 

 theorem. Mr Talbot's first theorem is a case of the fundamental 

 formula for the comparison of elliptic arcs. This remark has 

 reference to an inquiry which Mr Talbot suggests as to the 

 relation in which his theorems stand to the results obtained by 

 Legendre and others. 



In conclusion, it may be well to observe that Mr Talbot 

 has remarked that, apparently, a solution discovered by Fagnani 

 of a certain differential equation cannot be deduced from Abel's 

 theorem ; but as this solution may be easily derived from 

 the ordinary formula for the addition of elliptic integrals of 

 the first kind it is manifestly included in the theorem in 

 question. 



II. 



11. I now come to the history of researches into the pro- 

 perties of particular classes of algebraical transcendents. The 

 earliest, and still perhaps the most important class of these 



