242 ON THE RECENT PROGRESS OF ANALYSIS. 



tions, as the generalisation of it by Abel, or more properly 

 speaking, the theory of which Eider's result is an isolated frag- 

 ment, is the foundation of our knowledge of the higher trans- 

 cendents. We may therefore conveniently divide the subject 

 of this report into two portions, viz. the general theory of the 

 comparison of algebraical integrals, and the investigations which 

 are founded on it. Mathematicians have been led, by comparing 

 different transcendents, to introduce new functions into analysis, 

 and the theory of these functions has become an important 

 subject of research. 



The second portion may again be divided into two, viz. 

 the theory of elliptic functions, and that of the higher trans- 

 cendents. 



This classification, though not perhaps unexceptional, will, 

 I think, be found convenient. 



7. About sixteen years after the publication of Lagrange's 

 earlier researches on the 'comparison of algebraical integrals, he 

 gave, in the New Turin Memoirs for 1784 and 1785, a method 

 of approximating to the value of any integral of the form 



\~~fT ? where P is a rational function of. x and E the square 



root of a polynomial of the fourth degree. I shall consider this 

 important contribution to the theory of elliptic functions in con- 

 nexion with the writings of Legendre. At present, in order to 

 give a connected view of the first division of my subject, it will 

 be necessary to go on at once to the works of Abel, and to those 

 of subsequent writers. In the history of any branch of science 

 the chronological order must be subordinate to that which is 

 founded on the natural connexion of different parts of the 

 subject. 



I shall merely mention in passing, that in 1775, Landen 

 published in the Philosophical Transactions a very remarkable 

 theorem with respect to the arcs of a hyperbola. He showed 

 that any arc of a hyperbola is equal to the difference of two 

 elliptic arcs together with an algebraical quantity. In 1780 he 

 published his researches on this subject, in the first volume of 

 his Mathematical Memoirs, p. 23. This theorem, as Legendre 

 has remarked, might have led him to more important results. 

 It contains the germ of the general theory of transformation, the 



