ON THE RECENT PROGRESS OF ANALYSIS. 241 



function. With one form of assumed relation we are led to the 

 differentials of circular, and with another to those of elliptic 

 integrals, and so on. It is in this manner that Dr Gudermann, 

 in the elaborate researches which he has published in Crelle's 

 Journal, has commenced the discussion of the theory of elliptic 

 functions. 



5. In the fourth volume of the Turin Memoirs, Lagrange 

 accomplished the solution of the problem suggested by Euler. 

 He integrated the general differential equation already mentioned 

 by a most ingenious methoo, which, with certain modifications, 

 has remained ever since an essential element of the theory of 

 elliptic functions. He proceeded to consider the more general 

 equation 



dx _ dy 



where X and Y are any similar functions of x and y respectively, 

 and came to the conclusion, that if they are rational and integral 

 functions, the equation cannot, except in particular cases, be 

 integrated, if they contain higher powers than the fourth. He 

 also integrated this equation in a case in which X and Y involve 

 circular functions of the variables. It had been already pointed 

 out in the summary of Euler's researches, given in the Nov. 

 Com. Pet. Tom. vi., that if X and Fare polynomials of the sixth 

 degree, the last-written equation does not in general admit of an 

 algebraical integral, since, if so, it would follow that the solu- 



fJ/Y* // ?/ 



tion of the equation ' = - ^-5 , which (as the square of 



1 + x 3 is a polynomial of the sixth degree) is a particular case 

 of that which we are considering, could be reduced to an alge- 

 braical form. Now this solution involves both circular functions 

 and logarithms, and therefore the required reduction is impossible. 

 This acute remark* showed that Euler's result did not admit 

 of generalisation in the manner in which it was natural to 

 attempt to generalise it. It was reserved for Abel to discover 

 the direction in which generalisation is possible. 



6. The discovery of Euler, of which we have been speak- 

 ing, is in effect the foundation of the theory of elliptic func- 



* M. Richelot, in one of his memoirs on Abelian or [hyper- elliptic integrals, 

 quotea it, in a slightly modified form, from Eider's Opuscula. 



16 



