164 LAMBERT — EXPANSIONS OF ALGEBRAIC FUxVCTIONS. [April 7, 



confined to one fast disappearing pool, would be observed when 

 dotting the ground over an extent perhaps of an acre or more. 

 Seen thus, immediately after rain, and not previously noticed, the 

 inference is not so strange that they came to the earth with the 

 rain, or that there had been a shower of toads as well as of water. 

 Trenton, N./., April j, igo4. 



EXPANSIONS OF ALGEBRAIC FUNCTIONS AT 

 SINGULAR POINTS. 



BY PRESTON A. LAMBERT. 

 {Etad April 7, I904.) 

 I. Introduction. 



An algebraic equation F{x, y) = o of degree 7i \x\ y defines jf as 

 an ^-valued algebraic function of x. When these n values of y are 

 all distinct for a given value oi x, that value of ^ is called a regular 

 point of the algebraic function, and the 7t branches of the function 

 are extended by applying the law of the continuity of each branch. 



In curve tracing x and y are real variables and only the real 

 branches of the function are used. Real values of x and y which 



6F SF 



gatisfy the equations F(x,y) = o, y =zo, -r- = o determine mul- 

 tiple points of the curve which represents the equation T'{x, y) = o. 

 If ^ = ^, J' = ^ is a multiple point of this curve, the behavior of 

 the curve at the multiple point is determined from the expansions 

 oi y — b in terms of :r — a. Inasmuch as the transformations 



x = x^-\-a,y=y^-{- b 

 transfer the origin to the multiple point, the multiple point will 

 always be taken at the origin. 



An algebraic equation between complex variables F{w, z) = 

 of degree n in w defines w as an ^-valued algebraic function of z. 

 Values of w and z which satisfy the equations F{zv, s) = o and 



— F{w, z) = o, determine branch points of the algebraic func- 

 tion, that is points where several branches of the function meet. 

 The behavior of the function at a branch point is determined from 

 the expansions of the function at the branch point. 



