308 ON THE RECENT PROGRESS OF ANALYSIS. 



which introduce an independent variable t, symmetrically re- 

 lated to x, y and z ; and so in all cases. 



M. Richelot* then takes a symmetrical function of x, y, ...z, 

 viz. their sum, and by means of the last written equations arrives 

 at an integrable differential equation of the second order, the 

 principal variable being the said sum and the independent 

 variable being t. From the first integral of this equation it 

 is easy to eliminate the differentials, and we have thus an alge- 

 braical relation in x, y, ... z, from which, in the manner already 

 mentioned, M. Richelot deduces another. We now see that if 

 we could find any other symmetrical function which would lead 

 to an integrable equation we should get a finite relation among 

 the variables. 



In the next volume of Crelle's Journal M. Jacobi took the 

 following function as his principal variable, 



fju being a root of X = or fx if we suppose X =fx. Call- 

 ing this function v, we get a simple differential equation in v and 

 t, and a corresponding integral of the system. Now fx = 

 has 2m or 2m 1 roots, and we only want m 1 integrals. 

 The integrals therefore which we get by making p the first, 

 second, &c. root of fx = are not all independent. 



* As M. Richelot's method of demonstrating Euler's theorem is more sym- 

 metrical and far more easily remembered than Lagrange's, it ought, I think, to be 

 introduced into all elementary works on elliptic functions. The equation to be 

 integrated being 



, 



V a + fiy + yy* + Sy* + ey* 

 dx 





assume , 



dt y-x 



Then 



dt x-y 



Let p = x + y. Then after a few obvious reductions 



Hence 



- * a + 



the algebraical integral sought. It may easily be expressed in other forms. 



