286 ON THE RECENT PROGRESS OF ANALYSIS. 



done by what is called Abel's theorem, in at least two different 

 ways: the one, that of which he now makes use; the other, 

 that which we have seen is applied to the case of four functions 

 by Legendre in his third Supplement. 



He next determines the most general form of which the 

 integral of an algebraical differential expression of any number 

 of variables is capable, provided it can be expressed linearly 

 by elliptic integrals and logarithmic and algebraical functions. 

 The result at which he arrives admits of many important ap- 

 plications. It is, that the integral in question may be expressed 

 in a form in which the sine of the amplitude of each elliptic 

 integral and the corresponding A, and also the algebraical and 

 each logarithmic function are all rational functions of the varia- 

 bles and of the differential coefficients of the integral with re- 

 spect to each. 



He proceeds by an interesting train of reasoning to establish 

 the remarkable conclusion, that the general problem which we 

 are considering may ultimately be reduced to that of the trans- 

 formation of elliptic integrals of the first kind. The problem 

 of this transformation is then discussed, and by a method 

 essentially the same as that of which he had made use in his 

 paper in Schumacher's Journal. The appearance however of 

 the two investigations is dissimilar, because no reference is made 

 to elliptic functions (as distinguished from elliptic integrals) in 

 the first part of the Pr&cis. The relations therefore which 

 exist among the roots of y = tyx are established by considera- 

 tions independent of the periodicity of elliptic functions ; though 

 it is not difficult to perceive that they were suggested by the 

 results previously obtained by means of that fundamental pro- 

 perty. It is shown, that if the equation y tyx, where tfrx is a 

 rational function, satisfy the differential equation (A), then this 

 equation, considered as determining x in terms of y, is always 

 algebraically soluble. As the multiplication of elliptic integrals 

 may be considered a case of transformation (that, namely, in 

 which the modulus of the transformed integral remains un- 

 changed), this theorem may be looked on as an extension of 

 that which we have spoken of (p. 275) in giving an account of 

 Abel's first memoir on elliptic functions. The two theorems 

 are proved by the same kind of reasoning. 



The second part of the memoir was to have related to cases 



