172 LAMBERT — EXPANSIONS OF ALGEBRAIC FUNCTIONS. [April 7, 



Example II. — To illustrate the method of finding the successive 

 approximations let it be required to determine to three terms the 

 expansion of* the branches of the cycle corresponding to the under- 

 scored terms of the equation 



( 1 ) m' — z^w" — ^iv — ^^ =z o. 



Introducing a factor / into the terms of (i) which are not under- 

 scored, then differentiating twice with respect to / considering z 

 constant, 



(2) w' — ^ur — z'wt — ^ = o. 



/ \ i ^'"^ A dw 1 dw 7 in 



\\) ^"^ —r. — 2ZW-7: — 2/-— — z^w — s'" = o. 

 ^■^^ ^ di dt dt 



Making / = o in (2), (3), (4) 



(«,). = .', C^\ = i/ + i. % r^"j. = - |r 



Substituting in Maclaurin's series 



^=(^^)«+('^)»'+(^')»i 



and making /= i in the result we find 



(5) w^z^-\-\z^^\z^ 



which is correct to three terms. Equation (5) has the form of a 



power series in z^ beginning with the fourth power and represents a 

 cycle of three branches of the algebraic function w hanging 

 together at the singular point. 



Lehigh University, South Bethlehem, Pa., April y, igo4. 



