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



Suppose the algebraic equation when the singular point is taken 

 as^origin to have the form 



( I ) Gxy + j^xy + jix'y' + /xy + j^rxy + ^ 

 -f /x'y -I- Dxy H- Zxy + Cxy -f ^^^> -f ax^' = o, 



where^the terms are arranged according to the descending powers 

 o(y. In this equation y has fourteen branches, which are to be 

 separated at the singular point by expanding j; as a function of x. 



The terms of equation (i) to be underscored are determined by 

 the method used for this purpose in the paper on the ^' Solution of 

 Equations," and which is adapted to the present case as follows. 

 If Z^"^ij'"i, Mx'^^yn.^ JVx'^'ayr^s are any three terms of equation 

 (i), the value of 



(2) limit J/Pi-ng ^m2(°l-° 3) 



is zero, finite, or infinite. It is at once seen that this limit is zero, 

 finite, or infinite, according as WaC^^i — ^3) is greater than, equal to, 

 or less than ;;/i(??2 — ^h) + ^3(^1 — ^^2)- 



The underscored terms of equation (i) are all the terms 

 which satisfy the following condition. The limit (2) for any three 

 consecutive underscored single terms is infinite. If a group of 

 terms is underscored as a single term, the limit (2) is finite for all 

 the terms of this group, and the limit is infinite for the first term 

 of the group and the next preceding underscored term, the limit is 

 also infinite for the last term of the group and the next succeeding 

 underscored term. 



We now proceed to underscore the terms of equation (i) to 

 satisfy this condition. 



Underscore the first term of (i), and determine the limit (2) for 

 the first three terms of (i). Since the limit is infinite, underscore 

 the second term of (i) temporarily. 



Determine the limit (2) for the terms 2, 3, 4. The limit is 

 zero and term 3 is not underscored. Next determine the limit (2) 

 for the terms 2, 4, 5. The limit is infinite and term 4 is tempor- 

 arily, term 2 permanently underscored. 



The limit for terms 4, 5, 6 is zero, the limit for terms i, 2, 6 is 

 infinite. Hence term ['4 does not remain underscored, and term 6 

 is temporarily underscored. 



