ON THE RECENT PROGRESS OF ANAL YSIS. 243 



eccentricities of the two ellipses being connected by the modular 

 equation of transformations of the second order*. It is on this 

 account that in a report on M. Jacobi's Fundamenta Nova, 

 contained in the tenth volume of the Memoirs of the Institute, 

 Poisson speaks of Landen's theorem as the first step made in 

 the comparison of dissimilar elliptic integrals. Several writers 

 have accordingly given Landen's name to the transformation 

 commonly known as Lagrange's. 



8. We have seen that even Lagrange failed in obtaining 

 a result more general than that which had been made known by 

 Euler, and yet, as we now know, Euler's theorem is but a 

 particular case of a far more general proposition. But in order 

 to further progress, it was necessary to introduce a wholly new 

 idea. The resources of the integral calculus were apparently 

 exhausted; Abel, however, was enabled to pass on into new 

 fields of research, by bringing it into intimate connexion with 

 another branch of analysis, namely, the theory of equations. 

 The manner in which this was done shews that he was not 

 unworthy to follow in the path of Euler and of Lagrange. 



I shall attempt to state in a few words the fundamental idea 

 of Abel's method. 



Let us suppose that the variable # is a root of the algebraical 

 equation fa = 0, and that the coefficients of this equation are 

 rational functions of certain quantities a, b, ... c, which we shall 

 henceforth consider independent variables. Let us suppose also 

 that in virtue of this equation we can express certain irrational 

 functions f of x as rational functions of a?, a, b, ... c. For in- 

 stance, if the equation were x 2 + ax + - (a 2 1) = 0, it follows 



that Vl x' = a + x. So that any irrational function of the 

 form F (x Vl x*) can be expressed rationally (F being rational) 

 in x and a. 



* Vide infra, pp. 263 and 289. 



"h It must be remembered that an algebraical function is either explicit or 

 implicit : explicit, when it can be expressed by a combination of ordinary algebrai- 

 cal symbols ; implicit, when we can only define it by saying that it is a root of an 

 algebraical equation whose co-efficients are integral functions of x. Thus y is an 

 implicit function of a; if y 5 + xy+ i =o. The remarks in the text apply to all alge- 

 braical functions, explicit or implicit. 



162 



