ON THE RECENT PROGRESS OF ANALYSIS. 307 



Though in this memoir M. Richelot only obtained by direct 

 integration two of the m l algebraical integrals of the " Jaco- 

 bische system," yet he put the problem of its complete inte- 

 gration into a convenient and symmetrical form. As there are 

 m variables and m 1 relations among them, we may suppose 

 each to be a function of an independent variable t. Lagrange, 

 as we know, in integrating the equation 



cfo d L _ Q 



VZ VT ' 



introduced such an independent variable by the assumption 



dx 



which of course implied that -jf- = V Y. This assumption is 

 unsymmetrical, and it is therefore difficult to see how to gene- 



ralise it. But if we assume -r- = - , we shall of course have 



dt x y 



J/ - an d therefore t is symmetrically related to x and 

 dt y-x' 



y. Let Fu = be an equation whose roots are x and y, then, 

 as we know, when u x^ F'u~x y, and, when uy, 

 F'u = y x, so that using an abbreviated notation 



dx VT dy V? 



dt~F\x) a 1 dt~ F'(yY 



Nothing is easier than to generalise this result. For instance, 

 the " Jacobische system" of two equations is 



^L , dz A 

 + " 0> 



xdx ydy zdz 



Now if Fu have x, y, z for its roots, the two preceding 

 equations may, in virtue of a very well-known theorem, be re- 

 placed by the three following, 



_ _ ^ 



dt ~ F'x ' dt ~ Fy ' dt ~ F'z ' 



202 



