116 EHRMANN'S THEOREM. 



de Calcul Difffaentiel, 18me Lepon, and Liouville's Journal 

 de Mathdmatiques, torn. XI. p. 129 and 313. 



142. If x = a + y$ (x), we have by Lagrange's Theorem 

 * = + y 



where in the coefficients of the different powers of y, we are 

 to make x = a after the differentiations have been performed. 

 *y ___ ct 



Let y or = ^ (x), so that x = a is a root of ty (x) = ; 



9 ( x ) 

 then 



where, in the coefficients of 'the different powers of ty (x) after 

 the differentiations, x is to be made = a. This series for/ (a) 

 in powers of i/r (a;) is called Burmann's Theorem. 



143. Let ^(x) denote the inverse function of -fy (x), so that 

 if u = ty (x) we have ty' 1 (u} = x, and therefore ^ {ty~ 1 (u)} = u. 

 If we write ty~ l x for x in Burmann's Theorem, we have 



No change is made in the quantities in the square brackets, 

 for they do not contain x when the operations indicated are 

 completely performed. 



If f(u) = u, we have 



-a] a? d \(x a) 2 ~| 

 +-- f r / v 



L{f > ( aj )}J 







