244 ON THE RECENT PROGRESS OF ANALYSIS. 



From the given equation we deduce by differentiation the 



following, 



dx = ada + fidb + ... 



where cc, /3, ... 7 are rational in x, a, ,..., c. 



Let y be one of the functions which can be expressed ration- 

 ally in x, &c.j it follows that 



ydx = Ada +,Bdb + . . . + Cdc, 

 where A, B, . . . C are also rational in x, &c. 



The equation jfc = will have a number of roots, which we 

 shall call x^ x z , ... x^. It follows that 



da 



where the indices affixed to y, A, &c. correspond to those 

 affixed to x, so that y l9 for instance, is the same function of x l 

 that y z is of x 2 . 



Now A t + . . . + Ap is rational and symmetrical with respect 

 toajj ... a?^, therefore it can be expressed rationally in the co- 

 efficients of f(x) = 0, and therefore in a, b...c. We will call 

 this sum R a , and thus with a similar notation for b, &c. 

 we get 



The second side of this equation is from the nature of the case 

 a complete differential, and it is rational in a, b, c, &c.; it can 

 therefore be integrated by known methods ; and if we denote 



y i|r (ipj, we get 



M being a logarithmic and algebraic function of a, Z>, &c., which 

 we may suppose to include the constant of integration. 



^r (x) is in general a transcendental function, while a, b, &c. 

 are necessarily algebraical functions of x^ , . . . , x^ , and the result 

 at which we have arrived is therefore an exceedingly general 

 formula for the comparison of transcendental functions. 



The simplicity and generality of these considerations entitle 

 them to especial attention : it cannot be doubted that the ap- 

 plication thus made of the properties of algebraical equations to 



