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



IV. Behavior of Branches of Algebraic Functions. 



If the algebraic equation takes the form la^^x^f = o when a 

 regular point is taken as origin the method of expansion deter- 

 mines ;/ separate branches of the function. 



If the origin is a singular point of la^^x'^y'' = o the behavior of 

 the several branches is determined as follows. 



a) If the first approximation is independent of x or contains a 

 negative power of ^, the corresponding branches are either finite or 

 infinite and consequently these branches do not go through the 

 singular point. 



I?) To each pair of consecutive underscored terms in which the 

 exponents of y differ by unity there corresponds a separate branch 

 of the function through the singular point. 



c) To each pair of consecutive underscored terms in which the 

 exponents of y differ by more than unity there corresponds a sepa- 

 rate cycle of branches hanging together at the singular point, pro- 

 vided the exponents of x and y in the equation determining the 

 first approximation are prime to each other, and the number of 

 branches in the cycle equals the exponent of y. If, however, the 

 exponents of x and y in this equation have a common divisor 

 greater than unity, the corresponding branches break up into cycles 

 equal in number to the common divisor and the number of 

 branches in each cycle is the exponent ofy divided by the common 

 divisor. 



ti) If a group of terms is underscored and the equation formed 

 by equating this group to zero has equal roots, these equal roots 

 must be removed before the branches corresponding to the group 

 can be separated. If this equation is now solved the branches will 

 be separated into single branches and cycles of branches, provided 

 the exponents of y i'- this equation have no common divisor 

 greater than unity. Ii veer, there is a common divisor greater 

 than unity the branch'^'- ponding to this group break up into 



sub-cycles. 



V. A. , .:at.")ns in Curve Tracing. 



Example I. — Let it be required to trace the curve represented 

 by the equation 



(i) y — 2i^^y — x^y -)- (^x^y -\- 2x^ — 2^** = o 



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



