114 LAMBERT— SOLUTION OF ALGEBRAIC EQUATIONS [April 25. 



9. Lagrange's Theorem. — The equation 



a}'" + by^'x 4-^ = 

 may be written 



c b , 



(10) y = x-y\ 



Placing 3'" = z, whence y = ^^/", this equation becomes 



c h ^ 



ill) z= x-z'\ 



Lagrange's theorem asserts that if 



z = v^x<\>{z) 



/(,) =/{r) + x<i>{z.)/'{v) + ^^ % {Wff'^^')} + • • • 



+ ,!^{^(-)V'(^')}---. 



If now 



1 b ''- 



/iz)==z'-, c^(^)=-^-, 



and after the derivatives in series (12) have been formed z^ is replaced 

 by — c/a, there results, making x unity, 



■^ \ af nc\ at 2! ifr ^ ^ \ at 



(13) +317??^' + 3/^'- '0(1 + 3^'-2;0(-^) " 



b* i ^\ —- 



3 



In series (13) the law of formation of the successive terms is 

 evident and this law is readily proved by induction by using La- 

 grange's theorem. 



Series (13) may be more concisely written by placing 



so that ^0 is a root of the two-term equation 



