1904.] LA:\rBERT — EXPANSION'S OF ALGEBRAIC FCJNCTIOXS. 165 



The multiple points in curve tracing and the branch points in 

 algebraic equations between complex variables are grouped as the 

 singular points of algebraic functions. 



II. Historical. 



The problem of the expansion of algebraic functions at singular 

 points dates back to Newton. Newton's '* parallelogram method " 

 determines the first term of the expansions as follows. The equa- 

 tion transformed to the singular point as origin becomes 



Locate on squared paper to rectangular axes the points (-w, «) 

 whose coordinates are the exponents of x and y in the various 

 terms of the transformed equation. Connect by successive 

 straight lines, forming a broken line convex toward the origin, the 

 points nearest the origin. The sums of the terms of la^^x'^y'' = o, 

 for which the points (;;f, n) are located on the same straight line, 

 when equated to zero form equations which determine the first 

 terms of the expansions at the singular point. 



Puiseux in his classical *' Memoir on Algebraic Functions," 

 Liouville's Journal^ t. XV, 1850, used Newton's parallelogram 

 method and studied in detail the nature of the expansions of 

 algebraic functions. Puiseux' s Memoir is made the basis of Briot 

 and Bouquet's '' Elliptic Functions," and indeed is almost univer- 

 sally used in the study of algebraic functions. 



Nother's method in Arifiaien, IX, 1876, is representative of the 

 more recent methods of expansion of algebraic functions. By suc- 

 cessive quadratic transformations the singular point becomes a 

 regular point, and from the expansions at this regular point the 

 expansions at the original singular point are obtained by reversing 

 the quadratic transformations and the reversion of series. 



In the present paper an analytic method is presented which 

 determines not only the first terms of the expansions but also the 

 successive approximations of the several expansions. The method 

 of expansion used is that application of Maclaurin's series which 

 the author employed to compute all the roots of numerical equa- 

 tions and which is published in Vol. XLII of the Proceedings of the 

 American Pliilosophical Society. 



III. A New Method of Expansion. 



For convenience of description the exponents in the equation 

 to which the method is applied are assumed numerical. 



