1904.] LAMBERT — EXPANSIONS OF ALGEBRAIC FUNCTIONS. 169 

 Collecting terms in like powers of _y 



(2) / + (— s^* — ^' + 9^^') y + (2X' — 2X') = o. 



Since in a first approximation the lowest powers of x in the several 

 coefficients alone count, this equation may be written 



(3) / — z^*y + 2^« = o. 



The application of the method of underscored terms shows that 

 in equation (3) the three terms must be underscored as one term, 

 hence 



(4) / — 2>^'y + 2^^ = o. 



The equation / — ^xy -\- 2x^ = o has two roots each equal to 

 x\ 



Diminishing each root of equation (i) by x"^, if we write y =^ 

 J'l -f •'^^ we obtain the equation, 



(5) J^'l' + 3^'j'i' + (— ^' + 9^^') J'l + (— 3-^' -I- 9^-^') == o- 

 Retaining for a first approximation only the lowest powers of .r 



(6) yi' + 3^!>'i' — ^'yi — 3^' = o- 



In equation (6) the terms i, 2, 4 must be underscored, that is 



(7) yl + S^W — ^"^'yi — 3^ = o- 

 From equation (7) the first approximations oi y^ are 



(8) y^ = — 3.T^ ji = x\ y^ = — x\ 

 Consequently the first approximations of j' are 



(9) y =z — 2JC% y ^x"' -\ x^, y^x^ — x^. 



The three branches of y are separc :d by these approximations 

 and the behavior of the curve at the multiple point is found by 

 tracing the three equations (9) in the neighborhood of the origin. 



Exa7nple II. — Let it be required to trace the curve represented 

 by the equation 



(i) y — '^x'^y -\- (^x'y + 2x^ = o 

 in the neighborhood of the singular point (o, o). 



