ON THE RECENT PROGRESS OF ANALYSIS. 245 



the comparison of transcendents will always be a remarkable 

 point in the history of pure analysis. 



A very simple example may perhaps illustrate what has been 

 said. Let us recur to the equation 



- (a 2 -l)=0, ............... (1), 



and suppose that 



Differentiating the first of these equations, we find that 

 (2x + a] dx + (x + a) da = 0. 



Comparing this with the general expression of dx, we perceive 

 that 



and as 



, da 



^ 



so that 



Let # t and x 2 be the two roots of our equation, we have thus 

 to find the value of 



y = = ^ _^^ (vide ante, p. 243) ,* 



(2x, + a) (2# 2 + a) 

 since x 1 + x z = a. 



Hence & cfe x + y z dx z = 0, 



and \frx 1 + ^frx 2 = c. 



Since x t + x 2 = a, 



and ay 2 =-(a 2 -l), 



we see that x* + x* = l, or a? 2 = Vl a?^. 



* The ambiguous sign of the radical is to our purpose immaterial. 





