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

 To obtain a first approximation this equation may be written 



(2) y — 3*^t>'-}- 2^ =o- 



The three terms of this equation must be underscored as a single 

 term, when it is found that the equation from which the first 

 approximations are to be found has two roots each equal to jc". 



Transforming equation (2) by writing jv =}>i -{- ^% there results 



(3) yi + 3^W + 9-^>i + 9^' = o- 



In equation (3) terms i, 2, 4 must be underscored, which gives 



(4) ^'1' + S^'^W + 9xy,-^g^ = o. 

 The first approximations of )'i are 



(5) Ji = — 3^\ J'l = 3^'-^^ yi = — 3^*-^% 

 and consequently the first approximations of jf 



(6) y = — 2.T', y = x^ -\- ^I'x^, y = x'^ — y'x^. 



The approximations (6) separate the three branches of the 

 curve at the multiple point. 



VI. Applications in Functions of the Complex Variable. 



Example I. — Let it be required to determine the behavior of the 

 five-valued algebraic function defined by the equation 



at the branch-points of the function. 



The branch-points, the common solutions of (i) and the partial 

 derivative of (i) with respect to w, 



(2) ^w^ — 4(1 — 2^)70^=^0 



are located at 2; = o, ^ = =t i. 



At s = o the first approximations of w are determined by the 

 equation 



(3) ^'— ^'— J5!l=0- 



